# An Introduction to Crystallography and Diffraction

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```					Geometrical Crystallography
Part I: Two Dimensional Crystals

S. Asgari
Dept. of Materials Science & Engineering
Sharif University of Technology
Outline
 Motif, Lattice, Unit Cell

 Crystal System and Bravais Lattice

 Symmetry Operations in Two Dimensions

 The 10 Two Dimensional Point Groups

 The 17 Plane Groups
Motif, Lattice, Unit Cell
Crystal: A regular arrangement of atoms or
molecules repeated in one, two or three
dimension(s).

Crystal = Motif +Lattice

Motif: The repeat unit of a crystal.
Lattice: An imaginary network of points over
which motifs are distributed.
Motif, Lattice, Unit Cell
Unit Cell: A unit parallelogram of lattice which
contains the Motif.

Primitive Unit Cell : One lattice point per unit cell.
Two Dimensional Patterns
Lattice
An Important Property of Lattice:

The surroundings of all lattice points are identical.

Is this a lattice?
Example: Structure of Graphite
Each point represents a carbon atom.

Two carbon atoms per lattice point.
Lattice (Cont.)
Choice of Lattice (unit cell) is not unique.
Crystal System and Bravais Lattice

 Crystal System: Defines the shape of unit cell.

 Bravais Lattice : Defines the distribution of lattice
points within the unit cell.

Number of crystal systems and Bravais lattices in 2-D and 3-D

Crystal System Bravais Lattice
2-D Crystals               4              5
3-D Crystals               7             14
2-D Crystals

Crystal systems:                      Bravais Lattices:

Oblique (a ≠ b, ≠ 60°, 90°)          Oblique P
Rectangular (a ≠ b, =90°)            Rectangular P
Rectangular C
Square (a = b, =90°)
Square P
Hexagonal (a = b, =60°)
Hexagonal P

Note: (a = b, ≠ 60°, 90°) is sometimes called Rhombic or
Diamond but the suitable Bravais lattice is Rect. C.
2-D Bravais Lattices

Rectangular C and
Rhombic Lattices
Symmetry Operations
 An object or figure has symmetry if some
movement of the figure or operation on the figure
bring it to a position identical to that before the
operation.
 Symmetry operation of the first sort brings an
object to its identical shape, e.g. translation,
rotation and screw rotation.

 Symmetry operation of the second sort brings an
object to its enantiomorph (mirror or inverted
shape), e.g. reflection, inversion, glide reflection,
roto-inversion.
Rotational Symmetry

Symbols

Hermann-Mauguin
1, 2, 3, 4, 6

Schoenflies
C1, C2, C3, C4, C6
Limitation on Rotational Symmetry
Due to Lattice Regularity
- An isolated object may have n-fold rotation axis.
- A lattice may only have 1, 2, 3, 4, 6 fold rotation
axis.

b = 2 . a.cos  = m . a
Mirror Line

m
H
m

m
Lattice Symmetry
in 2-D Crystals

Note:

These are symmetry
elements of lattice
not crystal !

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