IMPORTANT

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! IMPORTANT !
Tuesday lectures have been re-located
to Ramphal R0.03/4
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Feedback

• Please give honest and critical feedback during
  the course
• During the lectures
• After the lectures
• Email t.p.a.hase_at_warwick.ac.uk

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The Muppets Guide to

• The Structure and Dynamics of Solids

2. How do we describe a Crystal? - Looking
towards Group Theory
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Unit Cell, Lattice and Basis
A crystal is a parallelepiped that is made up of
a regular repeat of some representative unit,
called the unit cell.
Unit Cell A volume of space bounded by lattice
points which describe the symmetry. It is defined
in terms of their axial lengths (a,b,c) and the
inter-axial angles (?,?,?).
TRANSLATIONAL SYMMETRY maps the unit cells across
the entire volume of the crystal
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A crystal

• The lattice forms a series of points in space
• The basis is a group of atoms placed on the
  lattice points

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Crystal Structure
Convolution of Basis and lattice
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Lattice and Basis
BASIS
LATTICE
CRYSTAL
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Lattice and Basis
BASIS
LATTICE
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Lattice and Basis
The basis can be convolved with the lattice in
different ways
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How Many Lattices are there?
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2D Bravais Lattices
A lattice is an infinite periodic set of points
defined by the three basis vectors, a,b and c.
In 2D total of 5 distinct lattices
T
Lattice vector
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Usually a unit cell is primitive i.e. it
contains one lattice point. More complex
structures are used to highlight the symmetry
Number of Lattice points in Primitive cell 1
Number of Lattice points in fcc cell 4
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Bravais Lattices 14 possible in 3D
F
I
P
I
P
P
F
I
C
P
P
C
P
P trigonal R- rhombohedral
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Other Symmetry Elements

• Introduction to Point Symmetry
• Rotation about an axis
• Reflection at a Plane
• Inversion through a point
• Rotoinversion

• 2 notations!
• Hermann-Mauguin or International system (used by
  crystallographers)
• Schoenflies System (Chemists)

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Rotational Symmetry

• An object has n-fold rotational symmetry if it is
  invariant under a rotation of

What are the allowed values of n?
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Crystallographic Constraints
Rotation about point 2 in row A takes 1 to the
position 1 in B
Similarly, rotation about (p-1) takes p to p
But these positions are all lattice points so
separated by a for appropriate rotation angles, a
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Crystallographic Constraints
In row A, points 1 and p separated by (m-1)a
where m is an integer In row B points 1 and p
separated by Xsa
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Crystallographic Constraints
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Crystallographic Constraints
As m and s are positive integers,
Therefore only n1, 2, 3, 4 or 6 allowed
International symbol 1, 2, 3, 4 or
6 Schoenflies Cn
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Mirror Planes
International symbol m Schoenflies is coupled
with rotations Cnh if mirror is normal to
rotation axis Cnv if mirror is parallel to
rotation axis
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Schoenflies mirrors
C2v
C2h
Mirror plane within the rotation axis
Mirror plane normal to the rotation axis
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Schoenflies
C1h or Cs
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Inversion through a point

• Inversion symmetry occurs if for each point
  (x,y,z) there is a corresponding point
  (-x,-y,-z), or

All Bravais lattices have inversion symmetry
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Point Symmetry of 2D Lattices
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Point Symmetry of 2D Lattices