MSE203
Structure and Characterization of Materials
Tutorial: 11-11:50 Thursdays
Submission: by 11 am on the corresponding Tutorial day
Assignment 3
Question 1
Graphene is a 2D allotrope of carbon. Carbon nanotubes (CNTs) are 1D allotropes of
carbon and three configurations, namely armchair, zigzag, and chiral configurations
of single wall carbon nanotubes are shown in the figure on the next slide.
Considering CNTs as pseudo 1D lattice, determine the lattice parameter and motif for
the three configurations if the side of the hexagonal carbon ring is “a”. You can use
the graphene sheet that is folded to obtain a CNT. [3 x 5 = 15]
Graphene
Armchair
Zigzag
Chiral
n = √𝟑𝒂
m = 2a
Answer 1
Graphene m Lattice parameter (side of the
rhombus) = √𝟑𝒂
Lattice points at centre of the ring
6-fold at the centre and 3-fold at
each carbon atom
Mirrors at 0, 30, 60, 90, 120, 150,
and 180 degree to horizontal
n = √𝟑𝒂
m = 2a
Armchair
Graphene
Lattice parameter = 𝒏 = √𝟑𝒂
Zigzag
Graphene
Lattice parameter = 𝟏. 𝟓 ∗ 𝒎 = 𝟑𝒂
Chiral Graphene
(Lattice parameter)2 =
(𝟓 ∗ √𝟑𝒂)𝟐 +(𝟑𝒂)𝟐 = 𝟖𝟒𝒂𝟐
Lattice parameter = 𝟗. 𝟏𝟔𝟓𝒂
5n + 1.5m
Question 2
Identify the plane group for the following and show all the symmetry elements in the
unit cell of the plane group [5 *4 = 20]
(a) (b)
(c) (d)
Please ignore the rhombic grid lines in patterns © and (d0
Solution 1 (a)
Plane group: pm
Two vertical mirrors
= +
Plane group: pm The motif is placed on a rectangular grid
with 2mm point group symmetry
The motif is the yellow house with red
background has a vertical mirror The pattern has a pm plane group
symmetry
The motif has “m” point group symmetry
Solution 2 (b)
Plane group: p4
Hatched squares
(not shown)
= +
The motif comprises of two squares which are
Plane group: p4 hatched with “4” point group symmetry
The motif is the yellow and red squares One at (0, 0) and another at (a/2, a/2)
sharing an edge with 45 degree rotation
with respect to the horizontal The pattern has a p4 plane group symmetry
Square unit cell
4 yellow squares with hatches at
corners and one red square with
hatches at center of square
If only red or yellow colour
Lattice is centered square which is
smaller square
Solution 2 (c)
Plane group: p6mm
The motif is a regular hexagon
The motif has a 6mm point group symmetry
The regular hexagon is placed on a rhombic or hexagonal lattice
with 6mm point group symmetry
The pattern has a p6mm plane group symmetry
Solution 2 (d)
m
Plane group: p3m1
The motif is a hexagon divided into 3 black and three white equilateral triangles
There are mirrors along the altitudes of the triangles
The motif has a 3m symmetry
The shaded hexagonal motif is placed on a rhombic or hexagonal lattice with
6mm point group symmetry
The pattern has a p3m1 plane group symmetry
Question 3
We have studied that there are only 5 unit cells possible in 2D and in the earlier
course on Introduction to Materials Science and Engineering you studied the 7 crystal
systems and 14 Bravais lattices. In this question,
1. Prove the absence of centered square lattice in 2D unit cell.
2. Find the 3D lattice obtained by stacking of the 2D square lattice (of dimension a)
▪ on top of each other at a distance a.
▪ on top of each other at a distance b such that b > a.
▪ on top of each other separated by a distance of a/2 and shifted by (a/2,
a/2).
3. Prove the absence of base centered cubic lattice.
[5 *3 = 15]
Answer 3
1. Prove the absence of centered square lattice in 2D unit cell
Centered square lattice is another square lattice
with dimensions of a/sqrt2
2. Find the 3D lattice obtained by stacking of the 2D square lattice (of dimension a)
• on top of each other at a distance a
• on top of each other at a distance b such that b > a
• on top of each other separated by a distance of a/2 and shifted by (a/2, a/2)
• on top of each other at a distance a → Simple Cubic
• on top of each other at a distance b such that b > a → Simple Tetragonal
• on top of each other separated by a distance of a/2 and shifted by (a/2, a/2) → Body
Centre Cubic
Answer 3
3. Prove the absence of base centered cubic lattice.
Base centred cubic lattice is equivalent to a
simple tetragonal with a c/a ratio of sqrt 2.
We also lose 3 fold axis along the body diagonal
as all the 6 faces are not the same.
Lattice is no longer cubic.
4 fold along the c axis is retained in the smaller
lattice so it is tetragonal.
