Problem $3($Graphene & CNT$)$

Please solve all questions and every part with explanation

A graphene sheet is a honeycomb lattice of carbon atoms $($see figure$).$ Let the distance between carbon atoms be $"$a$".$ A good model for graphene is to consider a single plane in which there is one valence electron per carbon atom. We will use the tight$-$binding approximation, in which this electron can occupy a single $p_z$ orbital at each carbon site. Let $R$ denote the centers of the hexagons in the honeycomb: these form the underlying hexagonal Bravais lattice. Please notice that the latter is indeed a Bravais lattice differently from the graphene honeycomb lattice. The unit cell spanned by $a_1$ and $a_2$ contains two carbon atoms conventionally labeled as A and $B$ atom, located at $R+v_A,R+v_B,$ as shown in the figure.

Carbon nanotubes are made up of a section of the graphene lattice that has been wrapped up into a cylinder. You can specify the way the lattice is wound up by identifying the winding vector $W.$ The winding vector must be a Bravais lattice vector, and so can be specified by two integers:

$W=n{a}_1+m{a}_2$

where $n$ and $m$ are integers. To construct a nanotube, take a graphene lattice and mark one atom $($either A type or B type$)$ as the origin. Shift the origin of the vector $W$ on the chosen atom. The new vector $W_n$ will point to another atom of the same type. Roll up the sheet perpendicular to $W_n$ so that the second atom sits exactly on top of the first. You have constructed a $(n$;$m)$ nanotube! Nomenclature: we can specify some special tubes said achiral: they are $(n$;$n)$ tubes which are called armchair tubes, and $(n$;$0$ zig$-$zag tubes. All other tubes are said chiral.

a$.$ Build a $(5$;$5)$ armchair tube $($i$.$e$.$ with scissors and adhesive tape!$)$ by making use of transparencies $-$ You can download this sheet from course homepage and print it out. The easy way to submit your homework is to take and print out a photo with finished structure on a white paper with you name on the paper as the evidence.

b$.$ Construct a $(8$;$0)$ zig$-$zag tube.

c$.$ Build a chiral $(7$;$3)$ tube.