CHAPTER 3: SYMMETRY
AND GROUPS, AND
CRYSTAL STRUCTURES
Sarah Lambart
RECAP CHAP. 2
„ 2
different types of close packing:
„  hcp
(ABABA)
„  ccp
(ABCABC)
„ Definitions:
74% occupied
CN = 12
octahedral & tetrahedral interstices
The coordination number or CN is the number of
closest neighbors of opposite charge around an ion. It can range
from 2 to 12 in ionic structures. These structures are called
coordination polyhedron.
RECAP CHAP. 2
Rx/Rz An ideal close-packing of sphere
for a given CN, can only be
achieved for a specific ratio of
ionic radii between the anions and
the cations.
1.0 1.0 - 0.732 C.N. Type Hexagonal or
12 Cubic
Closest Packing 8 Cubic 0.732 - 0.414 6 Octahedral 0.414 - 0.225 4 Tetrahedral (ex.:
SiO44-) 0.225 - 0.155 3 Triangular <0.155 2 Linear RECAP CHAP. 2
„ Pauling’s
rules:
„ #1:
the coordination polyhedron is defined by the ratio
Rcation/Ranion
„ #2:
The Electrostatic Valency (e.v.) Principle: ev = Z/CN
„ #3:
Shared edges and faces of coordination polyhedra
decreases the stability of the crystal.
„ #4:
In crystal with different cations, those of high valency and
small CN tend not to share polyhedral elements
„ #5:
The principle of parsimony: The number of different sites in a
crystal tends to be small.
CONTENT CHAP. 3 (3 LECTURES)
„ Definitions:
„ 7
unit cell and lattice
Crystal systems
„ 14
Bravais lattices
„ Element
of symmetry
CRYSTAL LATTICE IN TWO DIMENSIONS
„ A
crystal consists of atoms, molecules, or ions in a pattern that
repeats in three dimensions.
„ The
geometry of the repeating pattern of a
crystal can be described in terms of a
crystal lattice, constructed by connecting
equivalent points throughout the crystal.
„ Step1:
2D crystal lattice
CRYSTAL LATTICE IN TWO DIMENSIONS
„ Lattice
point: A crystal lattice is constructed by connecting
adjacent equivalent points (lattice points) throughout the crystal.
The environment about any lattice
point is identical to the environment
about any other lattice point.
„ 
„ The
choice of reference lattice point is
arbitrary.
One choice of a reference point
CRYSTAL LATTICE IN TWO DIMENSIONS
„ Lattice
point: A crystal lattice is constructed by connecting
adjacent equivalent points (lattice points) throughout the crystal.
„ The
basic parallelogram
(parallelepiped in three dimensions)
constructed by connecting lattice
points defines a unit cell.
A lattice constructed from the chosen point.
CRYSTAL LATTICE IN TWO DIMENSIONS
„ Lattice
Points and Unit Cell
Because the choice of reference lattice point is arbitrary, the
location of the lattice relative to the contents of the unit cell is
variable.
„ Regardless
of the reference point
chosen, the unit cell contains the same
number of atoms with the same
geometrical arrangement.
The same lattice and unit cell defined from a different reference point.
CRYSTAL LATTICE IN TWO DIMENSIONS
„ Unit
Cell
- The unit cell is the basic repeat unit
from which the entire crystal can be
built.
- A primitive unit cell contains only one
lattice point.
The same lattice and unit cell defined from a different reference point.
CRYSTAL LATTICE IN TWO DIMENSIONS
„ Alternate
lattice and choice of the unit cell
CRYSTAL LATTICE IN TWO DIMENSIONS
„  Rules
to choose a unit cell
„  Smallest
„  Highest
repeat unit
possible symmetry
(with the most 90° angles)
CRYSTAL LATTICE IN TWO DIMENSIONS
CRYSTAL LATTICE IN TWO DIMENSIONS
CRYSTAL LATTICE IN TWO DIMENSIONS
CRYSTAL LATTICE IN THREE DIMENSIONS
„ Unit
cell: The unit cell of a mineral is
the smallest divisible unit of a
mineral that possesses the
symmetry and properties of the
mineral.
The unit cell is defined by three axes or cell
edges, termed a, b, and c and three interaxial angles alpha, beta, and gamma,
such that alpha is the angle between b
and c, beta between a and c, and
gamma between a and b.
COORDINATION NUMBER VS. SITES
„ Example:
fcc structures
„ CN
= 12
COORDINATION NUMBER VS. SITES
„ Example:
fcc structures
„ CN
= 12
„ One
octahedral site
„ Two
tetrahedral sites
COORDINATION NUMBER VS. SITES
„ Example:
„  1)
fcc structures
Halite (NaCl)
„ Cl-:
ccp arrangment
„ Na+:
CN = 6: Na+ are in
octahedral sites
„ Cl-:
CN = 6: Cl- are in octahedral
sites
COORDINATION NUMBER VS. SITES
„ Example:
„  1)
fcc structures
Halite (NaCl)
„ Cl-:
ccp arrangment
„ Na+:
CN = 6: Na+ are in
octahedral sites
„ Cl-:
CN = 6: Cl- are in octahedral
sites
„ 2
ccp interpenetrating
„ Rock
salt structure
e.g.: LiF, LiCl, LiBr, LiI, NaF, NaBr,
NaI, KF, KCl, KBr, KI, RbF, RbCl,
RbBr, RbI, CsF, MgO, PbS, AgF,
AgCl, AgBr and ScN
COORDINATION NUMBER VS. SITES
„ Example:
„  2)
fcc structures
Antifluorite (Na2O)
„ O2-:
fcc arrangment
„ Na+:
in tetrahedral sites (CNNa =
„ Na+:
cubic arrangment
„ CNO
=8
4)
Rule: in any structure of
formula AxBy, CNA/CNB = y/x
Ex:. CNNa = 4 ➱ 4/CNB = ½➱
CNB = 8
COORDINATION NUMBER VS. SITES
„ Example:
„  2)
fcc structures
Sphalerite (ZnS)
„ S2-:
fcc arrangment
„ Zn2+:
in tetrahedral sites (CNZn =
4) but only one every other site
„ CNS
=4
Rule: in any structure of
formula AxBy, CNA/CNB = y/x
Ex:. CNZn = 4 ➱ 4 = CNS
COORDINATION NUMBER VS. SITES
„ Example:
Cesium Chloride (CsCl)
„ Cl-:
„ 
simple cubic arrangement
CNCs = 8
„ CNCl
=8
Rule: in any structure of
formula AxBy, CNA/CNB = y/x
Ex:. CNZn = 4 ➱ 4 = CNS
CRYSTAL LATTICE
„ Unit
cell in 3 D: 4 type of unit cells:
„  P:
„  I:
„  F:
primitive
Body-centered
Face-centered
„  C:
Side-centered
7 CRYSTAL SYSTEMS
c
c
c
a2
b
a
b
Triclinic
α≠ β≠ γ
a≠ b≠ c
Monoclinic
a α = γ = 90ο ≠ β
a≠ b ≠ c
c
c
a1
P or C
R
Hexagonal
Rhombohedral
α = β = 90ο γ = 120ο α = β = γ ≠ 90ο
a1 = a2 = a3
a1 = a2 ≠ c
a3
a2
b
a
Orthorhombic
α = β = γ = 90ο a ≠ b ≠ c
a2
a1 Tetragonal
α = β = γ = 90ο a1 = a2 ≠ c
a1
Isometric (or cubic)
α = β = γ = 90ο a1 = a2 = a3
14 BRAVAIS LATTICES
SYMMETRIES
“… is the hardest thing for student to understand, appreciate or visualize.”
SYMMETRIES
4-fold rotational symmetry.
SYMMETRY OPERATIONS
„ A
Symmetry operation is an operation on an object that results in
no change in the appearance of the object.
„ There
are 3 types of symmetry operations: rotation, reflection,
and inversion.
ROTATIONAL SYMMETRY
„ 1
fold rotation axis = no rotational symmetry
1
1
„ 2
fold rotation axis: identical after a
rotation of 180° (360/180 = 2)
symbol: filled oval or A2
2
2
2
ROTATIONAL SYMMETRY
„ 3
fold rotation axis = identical after a
rotation of 120° (360/120 = 3)
symbol: filled triangle or A3
fold rotation axis: identical after a
rotation of 90° (360/90 = 4)
symbol: filled square or A4
3
3
„ 4
4
4
ROTATIONAL SYMMETRY
„ 6
fold rotation axis = identical after a
rotation of 60° (360/60 = 6)
symbol: filled hexagon or A6
6
6
IMPROPER ROTATIONAL SYMMETRY
„ 5
fold, 7 fold, 8 fold or higher: does not exist in crystals because
cannot fill the space
MIRROR SYMMETRY
„ A
mirror plan is something that gives you the reflection that
exactly reflects the other side: same distance, same
component. The plane of the mirror is an element of symmetry
referred to as a mirror plane, and is symbolized with the letter
m.
MIRROR SYMMETRY
MIRROR SYMMETRY
MIRROR SYMMETRY
CENTER OF SYMMETRY
„ A
center of symmetry is an inversion through a point,
symbolized with the letter "i".
ROTOINVERSIONS
Combinations of rotation with a center
of symmetry.
„ 1 fold rotoinversion axis =
center of symmetry
symbol: A1
__
A1
ROTOINVERSIONS
1) Rotation 360°
2) Inversion
ROTOINVERSIONS
Combinations of rotation with a center
of symmetry.
„ 1 fold rotoinversion axis =
center of symmetry
symbol: A1
fold rotoinversion axis =
1) 180° rotation,
2) center of symmetry
= mirror perpendicular to the axis
symbol: A2
__
A1
„ 2
m
ROTOINVERSIONS
1) Rotation 180°
2) Inversion
ROTOINVERSIONS
„ 3
fold rotoinversion axis = 1) 120°
rotation, 2) center of symmetry
symbol: A3
„ 4
fold rotoinversion axis = 1) 90°
rotation, 2) center of symmetry
symbol: A4
1
ROTOINVERSIONS
„ 3
fold rotoinversion axis = 1) 120°
rotation, 2) center of symmetry
symbol: A3
1
„ 4
fold rotoinversion axis = 1) 90°
rotation, 2) center of symmetry
symbol: A4
ROTOINVERSIONS
„ 3
fold rotoinversion axis = 1) 120°
rotation, 2) center of symmetry
symbol: A3
fold rotoinversion axis = 1) 90°
rotation, 2) center of symmetry
symbol: A4
2
„ 4
2
ROTOINVERSIONS
„ 6
fold rotoinversion axis = 1) 60° rotation, 2)
center of symmetry = 3 fold rotation axis + 1
perpendicular mirror plan
symbol: A6
GLIDE AND SCREW
„ Glide
= translation + mirror
„ Screw=
translation + rotation
31
SYMMETRIES IN TWO DIMENSIONS
SYMMETRIES IN TWO DIMENSIONS