The figure below shows the structure of graphene $(2$D layer of carbon atoms$).$ The atoms labelled by A and B are both carbon atoms but their local environments are different.

$($a$)$ Identify the underlying $2$D Bravais lattice and the basis decorating each lattice point.

$($b$)$ Write the primitive vectors vec$(a{)}_1$ and vec$(a{)}_2$ shown in the above diagram in terms of the lattice constant $a($edge length of the parallelogram$)$ and the two unit vectors hat$(x)$ and hat$(y).$

$($c$)$ Given that the distance between the nearest neighboring carbon atoms is $1\mathrm{.}42.$ Find the value of the lattice constant $a.$

$($d$)$ The vectors vec$(a{)}_1$ and vec$(a{)}_2$ defines a parallelogram, which is the primitive unit cell. Find the area of the primitive unit cell.

$($e$)$ Using the given primitive vectors and introduce an additional vec$(a_3)=$hat$(z),$ which is a unit vector perpendicular to the plane of the system. Construct the vectors vec$(b_1),$vec$(b_2),$vec$(b_3).$ Use vec$(b_1)$ and vec$(b_2)$ as the primitive vectors, dot out the reciprocal lattice on a graph paper $($you can use the one provided on the last page$).$ Identify the lattice type of the reciprocal lattice $($it must belong to one of the $5$ possible Bravais lattice types in $2$D$).$

$($f$)$ Find the "area" of the primitive cell of the reciprocal lattice. How is this area related to the area of the primitive cell in the direct lattice, i$.$e$.$ the result in $($d$)?$

$($g$)$ Use another graph paper, dot out the direct lattice. Sketch a set of parallel crystal planes $($try some less trivial ones$).$ Identify the Miller's indices $($hk$)$ or $($hk$0)$ of these planes.

$($h$)$ On the graph of the reciprocal lattice in part $($e$),$ draw a reciprocal lattice vector vec$(G)(hk)=$ hvec$(b_1)+$kvec$(b_2)$ for the $($hk$)$ in part $($g$).$

$($i$)$ Don't need to write down anything for this part. Just do it privately. Put the graph paper with the direct lattice on top of the graph paper with the sketched reciprocal lattice $($look at them through strong light if you are using hard copies of the graph paper$),$ do you see that the reciprocal lattice vector vec$(G)(hk)$ in part $(h)$ is perpendicular to the set of planes you drew in part $(g)?$

$($j$)$ Use the graph of the reciprocal lattice. Take a lattice point and construct the first Brillouin zone.