Lecture 03 & 04:
Crystallography 2 & 3
Dr. Willem Kruger
Room 117
KrugerW@ukzn.ac.za
Recap questions
• Name 4 symmetry elements and operations we use to
describe the symmetry content of a crystal.
• Crystallography is the study of…
• Define “crystal”.
Inversion?
Mirror planes?
Rotational symmetry?
Inversion?
Mirror planes?
Rotational symmetry?
How many possible combinations can we make
with the different symmetry elements?
(a) Just 1
(b) Maybe around a dozen
(c) Guessing about 20-50
(d) Hundreds
Correct answer: 32
(e) Unlimited
Here they are in red:
They are called the
32 point groups or
crystal classes!
No need to memorize this table but…
The 32 point groups
are subdivided into
six crystal systems!
YOU MUST KNOW
THESE SYSTEMS AND
THEIR COMMON
SYMMETRY
ELEMENTS!
What is the subdivision into the
seven crystal systems based on?
How many point
groups exist for
crystalline substances?
Why is it not infinite?
To understand why, we need to
introduce the concept called the
Unit Cell
Unit cell definition
•A unit cell can be described as the smallest
possible unit of a crystal that contains all the
chemical components, structural features,
and symmetry of the crystal as a whole.
Name of this mineral?
CaF2
Unit cell of fluorite
Unit cell of fluorite
Unit cell of fluorite
Unit cell of fluorite
Each blue block on the left is
one of the above. The
stacking of the unit cell
creates a perfect cube.
Units cells can be stacked in three dimensions to
form a crystal lattice. A crystal lattice is defined as:
• Infinitely repeating pattern of atoms, ions,
or groups of atoms (molecules), to form a
crystal.
How many possible shapes of unit cells are
there?
• 14
• All minerals will fall into one of
these basic shapes, called
Bravais lattices.
• Unit cells are parallelepiped
• (a six-faced figure with each pair
of opposite faces parallel and of
equal size).
The 14 different unit cells
(Bravais Lattices)
P = Primitive
C = C-centred
I = Body-centred
F = Face-centred
The 14 different unit cells
(Bravais Lattices)
P = Primitive
C = C-centred
I = Body-centred
F = Face-centred
The 14 different unit cells
(Bravais Lattices)
P = Primitive
C = C-centred
I = Body-centred
F = Face-centred
The 14 different unit cells
(Bravais Lattices)
P = Primitive
C = C-centred
I = Body-centred
F = Face-centred
The 14 different unit cells
(Bravais Lattices)
P = Primitive
C = C-centred
I = Body-centred
F = Face-centred
120°
Quartz unit cell
Three of these unit cells will
make a hexagonal prism!
Repeating the hexagonal
prism will give you a
hexagonal quartz crystal!
What type of
unit cell is
this?
The 14 different Bravais Lattices
P = Primitive
C = C-centred
I = Body-centred
F = Face-centred
R = Rhombohedral
Bravais Lattices
determines the crystal
system.
For example, if a
mineral is constructed
with tetragonal unit
cells, it will belong to
the Tetragonal crystal
system.
What type of unit cell or Bravais lattice would a
crystal that looks like have?
How many point
groups exist for
crystalline substances?
Why is it not infinite?
It is possible to stack objects
with three-fold symmetry to
make a larger object with threefold symmetry. Same applies to
four-fold and six-fold rotational
symmetry.
Leads to gaps in
the crystal
structure!
Recap
• Define “unit cell”
• Define “crystal lattice”
• How many different types of unit cells are there?
• What do we mean when we say unit cells are parallelepiped?
• An isometric crystal will be composed of _________ unit cells.
• Explain how the unit cells in quartz is arranged to make a hexagonal
crystal.
• Explain why crystals with 5-fold rotational symmetry does not exist.
…But why exactly do we get crystal faces?
Why do unit cells tend to pack like this?
…and not like this?
Truncated octahedron
Why does quartz terminate in a bipyramid while beryl has a “flat” top?
How do crystal faces form?
• Crystals are made of building blocks called unit cells that contain all the structural and
chemical components and symmetry of the crystal as a whole.
• 14 types of unit cells exist (called Bravais lattices) that can be subdivided into trigonal,
monoclinic, orthorhombic, tetragonal, hexagonal, rhombohedral, and cubic.
• Stacking of these unit cells creates a crystal lattice. Depending on the exact type of
unit cell, the crystal that forms will fall into one of the seven crystal systems (e.g.
stacking of cubic unit cells will cause the crystal to crystallize in the isometric crystal
system). Stacking occurs in a regular way as to ensure a uniform charge distribution.
• To ensure a uniform charge distribution, crystals tend to develop flat faces.
• These flat faces usually arise along planes with the greatest density of atoms in the
crystal lattice.
• As a crystal grows, new atoms attach easily to the rougher and less stable parts of the
surface, but less easily to the flat, stable surfaces. Therefore, the flat surfaces tend to
grow larger and smoother, until the whole crystal surface consists of these plane
surfaces.
Formation of crystal faces (from Cairncross
and McCarthy)
“It can be seen that there are certain directions
in the grid along which the atoms are aligned
and closely spaced (f1 to f3 in figure 4.23A,
shown in previous slides and on the right).
Planar atomic arrangements such as these are
generally more stable than if the atoms were
heaped up in an irregular way because the
interatomic forces that bind atoms together in
the lattice are more uniformly distributed across
a planar surface. Under suitably stable and slowgrowing conditions, atoms will therefore tend to
form regular layers in this way, producing flat
surfaces. The orientations f1 and f2 in figure
4.23A have the closer spacing of atoms and will
be the most likely to form crystal faces (figure
4.23B). Along orientation f3, the atoms are
more widely spaced, so this orientation is
slightly less stable and hence occurs less
commonly.”
Unit cells don’t always stack to make a perfect
cube or crystal:
Recap:
• Define the term “unit cell”
• Define the term “crystal lattice”
• How many unit cell configurations exist?
• Explain, in your own words, how crystal faces develop. Refer to unit cells and
the crystal lattice in your explanation.
• Explain why crystals with 5-fold rotational symmetry does not exist.
Crystallographic systems
The classification is based on the relative
length and angles of the crystallographic
axes!
What exactly is a crystallographic axis?
A crystallographic axis is an imaginary line that
passes through the center of a crystal.
Three axes exist: the c-axis, which is upright,
and two (or sometimes three) horizonal axes, a
and b.
To see to which crystal system
a crystal belongs, imagine into
which shape of parallelepipe
it will fit.
Next, sketch in the three
crystallographic axis!
c
α
ß
a
b
γ
c
View of crystal
from top-down
α
ß
a
b
Classify according to length of axis and
angle between axis.
γ
a≠b≠c
α=ß=γ=90°
This is characteristic of the
orthorhombic crystal system!
c
α
ß
a
b
γ
Mineral example: olivine
Tetragonal crystal system
c
View of crystal
from top-down
α
ß
a1
γ
a2
a1=a2≠c
α=ß=γ=90°
Tetragonal crystal system
c
α
ß
a1
γ
a2
Mineral
example:
Zircon
Hexagonal crystal system
a1=a2=a3≠c
ß=90° γ=120°
Hexagonal system is subdivided into two
subdivisions:
• Trigonal
• Hexagonal
• The difference?
• Crystals in the
hexagonal division has
one 6-fold axis, while
in the trigonal division
it has one 3-fold axis.
Tourmaline (trigonal)
Beryl (hexagonal)
Isometric system
a3
a2
a1
a1=a2=a3
all axis at 90°
to each other
Isometric system
Pyrite
a3
a2
a1
Garnet
Diamond
Monoclinic
Diopside (clinopyroxene)
Triclinic
cc
b
a
a≠b≠c
α ≠ ß ≠ γ ≠ 90°
+c
You need to memorize all six systems and
remember the trigonal system as well! You
should be able to draw the images on the left
and indicate relative lengths of axis as well as
the angles between them.
Crystal systems are a product of the internal
structure of minerals and affect the physical
properties of minerals.
Learning the crystal systems is the first step so
you can learn to apply more advanced
techniques in mineral identification!
Once you have mastered this work, mineral
identification will become much easier!
Hardness along c-axis: 7
Along a-axis: 5
Along b-axis: 5-6
Recap
• Complete the sentence: based on their elements of
symmetry, crystals can be divided into 32 ......
• Complete: The 32 point groups are divided into 6 ……….
• Name all six
• Which crystal system has two subdivisions?
• Name the two subdivisions
• What is the main difference between the two subdivisions?
• Why do we care about studying all of this in the first place?
What crystal systems do the following
represent?
• a1=a2≠c
• α=ß=γ=90°
a≠b≠c
α ≠ ß ≠ γ ≠ 90°
a1=a2=a3
all axis at 90°
a1=a2=a3≠c
ß=90° γ=120°
What crystal system would you say this beauty belongs to?
This is orthopyroxene. Ortho = orthorhombic
This one?
This one?
Class attendance question
• To which crystal systems do the following crystals belong?
(a)
(c)
(b)