$($i$)$ Why is there an optical phonon branch in the dispersion relation of graphene, even

though it only contains one type of atom?

$($ii$)$ Graphene and monolayer Boron Nitride have identical structures, a $2$D hexagonal

lattice. What main difference would be expected in their dispersion relations? A

sketch may help to illustrate your answer.

iii$)$ The dispersion relation of a monatomic chain is described by the relation

$=\sqrt[\phantom{2}]{\frac{4C}{m}}|sin\frac{1}{2}ka|$

where $$ is the frequency, $k$ is the wavevector, $a$ is the spacing between the atoms,

$C$ is the spring constant of the bond between the atoms and $m$ is the mass of the

atoms. What is the maximum frequency that can propagate along the chain? Over

what range is this function fully defined, and what important feature of reciprocal

space does this range delineate?

iv$)$ An harmonic oscillator has $2$ degrees of freedom $($kinetic and potential energy$).$ In

the classical Dulong$-$Petit model what would be the total energy of a $3$D dielectric

solid of $N$ atoms? What would be the heat capacity $(C_v)?$

$($v$)$ Einstein assumed that the atoms in a dielectric solid behave as isolated harmonic

oscillators with quantised energies, oscillating at the same frequency. From these

assumptions show that the heat capacity in a $3$D solid is

$C_v=3N{k}_B\frac{x^2{e}^x}{(e^x-1{)}^2}$

where $x=\frac{}{K_{B}T},$ and $N$ is the number of atoms.

Also show that in the high temperature limit the Dulong$-$Petit value is recovered.