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LECTURE 8.1
Lecture Outline
Weekly deadlines
 Symmetry

Name
Structure 3: Bonding , Crystal Structures and Propertie s
Lectures
Lecture 8.1 Symm etry of Patterns
Lecture 8.2 Recogn izing Symm etry
Requi red Book Reading 1
(For the end-of-UNIT qui z)
Part D:
Chapters 22, 23, and 24,
plus the Epil ogue to Part D
Animations
Ian H arrison: "PolymerSyn thF"
“StateMatterF”
Biographies
Mineral of the Week
Assignments due this week
Assignment 5, A Materia ls' Essay; Part III, is due
Wednesday mi dnigh t
Prac tice Quiz
Practice qui z que stions are ava il able on ANGEL, in the
Lesson 08 folders.
End of Unit Quiz
Quiz 8 will cons ist of ~ twenty (20) que stions for a total of
fifty (50) points. Quizze s are “individua li zed” , but with the
que stion s taken from a large da tabase.
Material cove red: Book Reading and Anim ations
OUTLINE

Symmetry and the Crystalline State
 Mirror
Symmetry
 Center of Symmetry
 Rotational Symmetry
 Translational Symmetry
 M. C. Escher
 Symmetry of Graphite
MIRROR SYMMETRY

Mirror symmetry reflects left to right.
 There is a vertical mirror line, midway between
the two eyes
MIRROR SYMMETRY

Mirror symmetry reflects left to right.
 There is a vertical mirror line (m), midway
between the two hands
CENTER OF SYMMETRY

A center of symmetry
“inverts through a
point”.
 A hand, with thumb
“up”, and “palm-side”
facing out.
Transforms to a hand
with thumb “down”
and back-side facing
out.
ROTATIONAL SYMMETRY
The “motif” or pattern
is repeated every
360/n˚, where n = 1,
2, 3, 4,…
 To the left, n = 3, and
the motif (hand) is
reproduced every
120˚
 The symbol for the
“triad” axis is an
equilateral triangle.

FIVE-FOLD ROTATIONAL SYMMETRY
IN THE PLANT KINGDOM

CACTUS PLANT
 The “motif” (a petal)
repeats every 360/5 =
72˚.
 There is a “five-fold
(pentad)” rotation axis
perpendicular to the
flower-head.
FIVE-FOLD ROTATIONAL SYMMETRY
IN THE PLANT KINGDOM

MORNING GLORY
 The “motif” (a petal)
repeats every 360/5 =
72˚.
 There is a “five-fold
(pentad)” rotation axis
perpendicular to the
flower-head.
TRANSLATIONAL SYMMETRY


All crystalline materials
must display
“TRANSLATIONAL
SYMMETRY”, or:
All crystals must be
describable in terms of a
lattice which consists of
an infinite number of
lattice points, each of
which has identical
surroundings.
TRANSLATIONAL SYMMETRY





The lattice is described by a “unit
cell”, which defines the pattern
shape.
In two-dimensions, the unit cell is
described by two lengths, a and b,
and an angle, g
In two-dimensions, there are
five pattern shapes.
In three-dimensions, three
lengths (lattice parameters) and
three angles define the unit cell.
In three-dimensions, there are
fourteen pattern shapes
THE FIVE PLANAR PATTERNS





a) Parallelogram (can always
be found. Only used if no other
pattern can be used)
b) Rectangle
c) Centered Rectangle (always
preferred to a parallelogram)
d) Square (the two sides must
be equal in length, and the two
interior angles must be 90˚
Always preferred to a centered
rectangle)
e) Hexagon (all six sides must
be equal in length, and all
interior angles must be 120˚
Always preferred to a centered
rectangle)
SYMMETRY OF THE CLOSEST PACKED
PLANE IN BCC METALS
MIRROR SYMMETRY

The atoms are in
contact along the two
red lines.
 The angle ABC =
70˚32’
 AB and BC are not
mirror lines
 The horizontal and
vertical lines are
mirror lines
TWO-DIMENSIONAL LATTICES

a) is possible: indeed it is always possible to describe any planar pattern by
a parallelogram. However, it should only be used if no other pattern shape is
applicable.
 b) is the conventional pattern shape. The centered rectangle better reflects
the “symmetry” of the pattern, than does the parallelogram.
 c) is incorrect. The “hexagon” is not regular!
AN ESCHER PRINT
ESCHER PRINT II
ESCHER PRINT III
ESCHER PRINT IV: SQUARE
ESCHER PRINT IV: SQUARE
ESCHER PRINT V: HEXAGON
ESCHER PRINT V: HEXAGON
ESCHER PRINT VI:
RECTANGLE
ESCHER PRINT VI:
RECTANGLE
THE STRUCTURE, LATTICE AND
SYMMETRY OF GRAPHITE I

In a), a two-dimensional sheet of carbon atoms, in the graphite
structure is shown. The structure consists of a hexagonal distribution of
carbon atoms. A possible lattice site is indicated by the asterisk at A.
 If A is taken to be the original lattice site, then all such positions are
also lattice sites as shown in b).
THE STRUCTURE, LATTICE AND
SYMMETRY OF GRAPHITE II

In c), the lattice is outlined by the broken lines. The pattern shape is
hexagonal.
 In d), the rotational symmetries associated with points A, B, and C on a) are
shown to be six-fold, three-fold, and two-fold respectively. The mirror lines
that are associated with the three positions are also indicated.