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Slide 2 Lecture 1
Lattice structure: lattice with a basis.
Lattice structures of common chemical elements.
Concept of Bravais lattice, definition and examples.
Primitive vectors of Bravais lattice.
Coordination number.
Primitive/Conventional unit cell.
Wigner-Seitz primitive cell.
Examples of common crystal structures.
Body-centered cubic lattice.
Face-centered cubic lattice.
Crystal systems
Reciprocal lattice.
First Brillouin zone.
Lattice planes and Miller indices.
Syllabus
Slide 3 Lecture 1
Determination of crystal structures by X-ray diffraction.
Bragg formulation of X-ray diffraction by a crystal.
Von Laue formulation of X-ray diffraction by a crystal.
Equivalence of Bragg and Von Laue formulations.
Diffractometers
Symmetry, elements of point groups
Geometrical structure factor.
Atomic form factor.
Crystal defects.
Points defects
Line defects
Stress and Strain.
Syllabus
Slide 4 Lecture 1
Solid State Physics ~ Ashcroft & Mermin, [Holt-Saunders]
• A great text for anyone with an interest in the subject.
Solid State Physics ~ Hook & Hall, [Wiley]
• Useful text. Read as a compliment to Ashcroft or Elliott.
Introduction To Solid State Physics ~ Kittel, [Wiley]
• Covers a huge amount in basic detail.
The Physics and Chemistry of Solids ~ Elliott, [Wiley]
• Lateral reading. Quite Chemistry based. Good for problem solving.
Structure of materials ~ De Graef, McHenry,
[Cambridge]
• Covers a huge amount in basic detail, good for problem solving.
Recommended Reading
Slide 5 Lecture 1
Crystals
Solids can be categorised into either crystalline or non-
crystalline solids.
This course deals with the structures found in crystalline
solids i.e. crystals.
A Crystal is a solid in which the
constituent atoms, molecules,
or ions are packed in a
regularly ordered, repeating
pattern extending in all three
spatial dimensions.
Gallium Crystal
Slide 6 Lecture 1
Despite an underlying crystalline
structure the crystal itself may
not appear regular in shape.
A closer look at the
substance reveals a
repeating pattern.
Crystals
Slide 7 Lecture 1
Silicon (Si)
Examples
Principle component of
most semiconductors
Structure
Diamond Cubic.
More on that later
Slide 8 Lecture 1
Crystalline SiO2
Silicon Dioxide or Silica with a
definite crystalline structure.
Structure
Each Silicon atom is surrounded
by four Oxygen atoms.
Silicon Dioxide is an example of
Tetrahedral oxygen termination.
Oxidation State Electron
Configuration
O -2 [He].2s2.2p
6
Si 4 [Ne]
Examples
Slide 9 Lecture 1
Structure
As with the crystalline SiO2
- Most silicon atoms have 4
bonds.
- Most oxygen atoms have 2
bonds.
Amorphous SiO2
(Silica gel)
Silica gel is an example of a non-
crystalline solid.
The local symmetry is the same
as the crystalline SiO2
However, translational symmetry
is missing.
Examples
Slide 10 Lecture 1
Oxygen Termination
The two forms of oxygen termination are Tetrahedral and
Octahedral
These refer to the structure of the oxide in relation to the original
structure of the crystal.
Tetrahedral Oxygen termination
is when four oxygen atoms
create a tetrahedron around the
original atom e.g. Silicon
Octahedral Oxygen termination
is when six oxygen atoms create
a octahedron around the original
atom e.g. Aluminium
Slide 11 Lecture 1
Oxidation
State
Electron
Configuration
O -2 [He].2s2.2p
6
Al 3 [Ne]
Aluminium Oxide (Al2O
3)
Structure
Each Aluminium atom is
surrounded by six Oxygen
atoms.
Aluminium Oxide is an
example of Octahedral oxygen
termination.
Examples
Slide 12 Lecture 1
Magnesium Oxide (MgO)
Oxidation State Electron Configuration
O -2 [He].2s2.2p
6
Mg 2 [Ne]
Structure
Structure is the same as
that of Sodium Chloride.
i.e. F.C.C. lattice with a two
point basis
Again more on that later
Examples
Slide 13 Lecture 1
32 oxygen anions form an F.C.C. lattice
8 tetrahedral interstices are occupied
by Fe3+
ions
Magnetite ( Fe3O
4)
16 octahedral interstices are occupied by
Fe3+
and Fe2+
ions in equal proportions
This is an example
of Spinel structure.
Lattice constant
(Basic repeat unit)
a = 0.8397nm
Examples
a a
Slide 14 Lecture 1
Spinel Group
Refers to a class of minerals with the general formulation
A2+
B2
3+O
4
2-
The oxide anions arrange in a cubic lattice with the A and B
cations occupying the Tetrahedral and Octahedral sites.
Possibilities for A and B include Magnesium, Zinc, Iron,
Manganese, Aluminium, Chromium, Titanium and Silicon.
In the case of Magnetite (Fe3O
4) the Iron is both A and B. That is,
A and B are the same metal under different charges.
Fe3+
(Fe2+
Fe3+
)O4
2-
Slide 15 Lecture 1
High Performance Materials
The majority of the worlds electricity supply is generated in
power stations using steam turbines.
Through the use of coal,
nuclear power, etc.
steam is generated and
is passed through the
giant steam turbines.
The turbines rotate and
generate electricity.
2
11
T
T
Assuming the system can be considered
a Carnot Engine and using current
average values of temperatures in the
system (T1=35°C and T
2=540°C) the
efficiency is calculated to be 62%.
Slide 16 Lecture 1
High Performance Materials
As supplies of fossil fuels are
diminishing, there is large interest
in making the steam turbine a
more efficient process for
generating electricity.
A way to increase this efficiency
is to operate the system with a
larger temperature gap, ie by
making T2
larger.
Slide 17 Lecture 1
High Performance Materials
2
11
T
T
This is not as simple as it sounds. To
increase the efficiency by just 5% would
require an average operating temperature
increase from 540°C to 660°C.
The turbine blades themselves must be able
to withstand these high temperatures
without melting or buckling.
This is where knowledge of the crystal structure of
the materials used in their production is
invaluable. Without a detailed analysis of the
structure engineers and scientists would not be
able to combat the problem of creating a more
efficient system.
Slide 18 Lecture 1
Lattice
A Lattice is a regular, periodic
array of points throughout an area
(2D) or a space (3D).
This picture shows one of many
possible lattice types.
When dealing with crystal structures it is best to firstly
consider the mathematical idea of a lattice, without the
notion of atoms or molecules at this stage.
Example of a 2D Lattice
IMPORTANT NOTE: The lattice is the underlying pattern of
the crystal. The crystal is being described by the lattice
that can contain more than one atom/ion assigned to
each point of the lattice. This is called a Basis.
Slide 19 Lecture 1
Basis
You have a lattice
with more than one
atom/ion assigned to
each lattice point.
Atoms can be the
same or different.
Complex structures
will have larger/more
complex bases.
All crystal structure
consists of identical
copies of the same
physical unit, called
the basis, assigned
to all the points of
the lattice.
Consider a point of the lattice
Now introduce a arbitrary basis of two atomsThen apply to the rest of the lattice
Lattice + Basis = Crystal
Slide 20 Lecture 1
While the basis is
assigned to each point
of the lattice there is
nothing to say that it is
anywhere near this
point.
Consider the same
lattice and basis as
before.
Basis
Increase the separation
between the basis and
the lattice point.
We discover that the
same lattice is revealed
when the new position
of the basis is applied
to each lattice point.
Slide 21 Lecture 1
However, there is no
unique choice of Basis!
Even the most complex
crystal structures can be
broken down into a
lattice and a basis.
Basis
Lattice + Basis =
Crystal
Each of these Basis,
when combined with
this lattice, will yield
the same crystal.
Slide 22 Lecture 1
Symmetry of a Basis
Consider a Basis that is itself symmetric and combine with the
lattice to create a simple crystal structure;
Crystal properties
considered along
the x-axis are
independent of
direction (left or
right)…
… they are also
independent of
direction along the
y-axis (up or
down).
Slide 23 Lecture 1
Now consider a basis that is not completely symmetric;
Crystal properties
considered along
the x-axis are
independent of
direction (left or
right)…
… However the
structure now has a
lower symmetry
than before and is
not the same from
the top and
bottom.
Symmetry of a Basis
Slide 24 Lecture 1
What if the basis is completely asymmetric?
The properties of
the substance may
no longer be
independent of
direction they are
measured.
Symmetry of a Basis
Slide 25 Lecture 1
Bravais Lattice
The foundation of ANY Crystal structure is the Bravais lattice.
Definitions of Bravais Lattice
The vectors used to define a Bravais lattice,
a1, a
2, a
3 ,are called the primitive vectors.
1. Infinite array of discrete points that appear exactly
the same from whichever of the points the array is viewed.
2. All the points with position vectors
R = n1a
1+ n
2a
2+ n
3a
3
Where; a1, a
2, a
3 are three vectors not all in the same plane
and n1, n
2 , n
3 are integer values.
3. A discrete set of vectors, not all in the same plane, closed
under vector addition or subtraction.
Slide 26 Lecture 1
One of the most common three-
dimensional Cubic Bravais
lattices, the Simple Cubic lattice.
All a1, a
2, a
3 ,are of equal length
and orthogonal.
a1
a2
a3
Pictured: A general two-dimensional
Bravais lattice of no particular symmetry.
The vectors a1
and a2
are primitive
vectors.
Bravais Lattice
Slide 27 Lecture 1
Non-Bravais Lattices
Using the first definition of a Bravais lattice
it is clear that the 2D Honeycomb Structure
is not a Bravais lattice.
The lattice does not appear exactly same
when viewed from P, Q and RP R
Q
In 3D an example of a non-
Bravais lattice is the Hexagonal
Close-Packed structure.
The points of the middle layer
are above the centres of the
triangles in the layer below. E.g.
if you double the vector from
corner point to centre point, you
will NOT arrive to another point
of the lattice.
Slide 28 Lecture 1
2D Non-Bravais Lattice with a Basis
All Non-Bravais lattices
can be created from a
Bravais lattice with a
basis.
The red points form a
simple 2D bravais lattice.
Adding a two-point basis
to each lattice point…
and it starts to look
familiar.
So if we go back to the case of Honeycomb structure. While it
itself is not a Bravais lattice, it can be considered to be a Bravais
lattice with a two-point basis.
Slide 29 Lecture 1
As mentioned before, the choice of basis is not unique, there are
infinitely many bases to choose from. However, that does not
mean all combinations of two points will work…
Consider a lattice point
and the basis shown.
Apply this basis to the
same lattice as before.
If we super-impose the
picture of the honeycomb
over ours it is obvious
that they do not match.
2D Non-Bravais Lattice with a Basis
Slide 30 Lecture 1
In each of the following cases indicate whether the structure is a
Bravais lattice.
a) Base-centered Cubic (Simple cubic with additional points in the
centres of the horizontal faces [2])
b) Side-centered Cubic (Simple cubic with additional points in the
centres of the vertical faces [4])
c) Edge-centered Cubic (Simple cubic with additional points at the
midpoints of the lines joining the simple cubic points [12])
Exercise
If it is a Bravais lattice give three primitive vectors.
If not describe it as Bravais with the smallest possible basis.
a) b) c)
Slide 31 Lecture 1
Questions/Problems
What is a Crystal?
What is a Bravais Lattice?
What is a Basis?
What is meant by the Symmetry of a basis?
I would urge you to know the answers to these questions before
next time.
Good resources
Solid State Physics ~ Ashcroft, Ch. 4
Introduction to Solid State Physics ~ Kittel, Ch. 1
The Physics and Chemistry of Solids ~ Elliott, Ch. 2
Solid State Physics ~ Hook & Hall, Ch. 1