\section{Problems}

\begin{enumerate}
\renewcommand{\labelenumi}{ [\arabic{chapter}-\arabic{enumi}] }
\item
Obtain the frequencies of two atoms (of mass M) which are connected
to each other by a
spring with spring constant K  and also connected to the two walls (that
is, wall-atom-atom-wall geometry). Here we consider only the longitudinal
modes for  which the vibration is along the bond axis. We consider that
the walls have an infinite mass. Show the  two normal modes in a
figure, using arrows  to show the atomic mode displacements.

\item
Obtain the vibrational frequencies of three atoms in similar
geometries to the previous problem, which involves the
wall-atom-atom-atom-wall geometry. Show the three normal modes using arrows in
the figure to denote atom displacements. In order to solve this problem,
you can use the fact that any normal mode should be either symmetric
or anti-symmetric under the inversion operation $x \to -x$.

\item
Next we consider $N-1$ atoms attached between the two walls.  When we
consider $x_0$ and $x_n$ as the coordinates of the two walls, show
that all the equations of motion are expressed by the same formula.
Then considering that the Bloch theorem applies, substitute $x_\ell =
A \exp(iq\ell a - i\omega t)$ into the equations of motion and obtain
the dispersion of the phonon frequency $\omega(q)$. Plot the
dispersion of the phonon within the first Brillouin zone.

\item
In the previous problem [3-3], we can use the fixed boundary condition
of $x_0 = x_n = 0$. Obtain the $N-1$ $q$-independent phonon frequency
values from the boundary condition.

\item
What happens in the previous problem [3-3], if we adopt the periodic
boundary condition $x_0 = x_n$. Here we consider the center of
mass motion to be zero.

\item
Show that the previous results satisfy the case of $N-1 = 2$ and $N-1=3$
by comparing your results with answers to earlier questions in Chapter 3.

\item
The problems above describe the phonon dispersion for a one-atom per unit cell
1D crystal. If you replace each even atom by a different atom, the system
will become a two-atoms per unit cell 1D crystal, with a doubled-size unit cell.
Show that the first Brillouin zone is reduced to half of the size of the
original one and the phonon dispersion will be represented by a zone
folding of the one-atom phonon dispersion.

\item
In Fig.\,\ref{f-COcrystphon}, choose any unit cell vibrational mode and
show that the motion of the atoms for $q = 0$ ($\lambda \rightarrow
\infty$) and for $q = 2\pi/a$ ($\lambda = a$) is the same.

\item Consider the phonon dispersion of a two-dimensional square
lattice with atoms of mass M and spring constant K. Plot the resulting
phonon dispersion curve as a function of $k_x$ and $k_y$.

\item Consider the phonon dispersion of a two-dimensional honeycomb
lattice with atoms of mass M and spring constant K. In this case, we
have two atoms per unit cell. Show the first Brillouin zone and
plot the phonon dispersion for the high symmetry directions within the
first Brillouin zone.

\item
Consider a $2(N-1)$ linear chain of atoms in which two different
atoms A and B with masses $M_A$ and $M_B$ are connected along a chain
(wall-A-B-A-B-... -B-wall). Obtain and plot the phonon
dispersion for this configuration.

\item
In the previous problem, show the normal modes for the $\Gamma$ point
and for the zone boundary for each phonon mode.

\item
Consider the $2(N-1)$ linear chain with one type of atom (mass $M$) with
two different spring constants alternating in the sequence
wall-(K1)-atom-(K2)- ... -(K1)-wall.  In this case, we have two phonon
branches. Plot the phonon dispersion.

\item
When we consider a transverse phonon mode, how should we consider the
force constant for a linear chain of similar atoms of mass M. Show
that stretching a spring in such a linear chain does not give a
deformation which is proportional to $y$ or $z$ when we consider the
direction along the chain as $x$.

\item
How many normal modes exist for the CH$_4$, C$_2$H$_2$ and C$_{60}$
molecules which, respectively, have the shapes of
a regular tetrahedron, a linear
chain, and a regular icosahedron?

\item
Let us consider the H$_2$O molecule. Solve for the phonon normal modes
by considering the spring constant $K_1$ for the H-O bond stretching force
constant and $K_2$ for the H-O-H bond angle force constant. Obtain
these force constants by using the experimental values for the mode
frequencies.

\item
When we consider the two dimensional square lattice for atoms of mass
M and force constant $K$ between nearest neighbor atoms, write an
equation of motion for the transverse phonon mode and give a solution
for its mode frequency.

\item
Let us consider the phonon modes of the C$_{60}$ molecule.
When we consider two spring constants for pentagonal
and hexagonal C-C bonds, show how to construct the dynamical matrix
and how to calculate the phonon modes.

\item
Consider zigzag and armchair graphene nano-ribbons which have
edges with zigzag and armchair shapes. Obtain discrete $k$ vectors in
the nano ribbon width direction by defining the width of
the nano-ribbon.

\item
Consider a ring which consists of $N$ carbon atoms, each connected to
its neighboring atoms by spring constant $K$ and consider only the radial
breathing phonon mode. Then show that the corresponding phonon frequency is
inversely proportional to $N$.

\item
When we consider the hexagonal corner of the two dimensional Brillouin
zone of graphene, that is the $K$ and $K'$ points, show that the phonon
eigenvectors have the periodicity of a $\sqrt{3}\times\sqrt{3}$
super-cell.  How many atoms exist in this super-cell?

\item
Consider the previous problem [3-21] for the case of the M point,
which is the center of the hexagonal edge of the two dimensional
Brillouin zone.

\item
Study the general theory of the LO and TO phonon modes whose mode
frequency ratio depends on the dielectric constant of the materials.  In
particular, show that the LO and in plane TO phonons are degenerate at
the $\Gamma$ point ($q=0$) in graphite.

\item
The sound velocity of the LA phonon mode of graphite is about 21km/s.
Estimate the spring constant of the C-C chemical bond of graphite.

\item
Review the principles of inelastic neutron scattering and inelastic
x-ray scattering. What is the merit of each of these experimental techniques
for studying carbon systems?

\item
In order to get momentum and energy information from inelastic neutron
scattering measurements, we need a monochromator to disperse the neutron
beams. How do we get a neutron beam with a fixed kinetic energy?

\item
When we use 10\,KeV x-rays for observing 0.1\,eV phonons, we need high
accuracy for observing the scattering angles. Estimate the accuracy of
the angles needed for the x-ray detector in such an experiment.

\end{enumerate}
