Done, your profile is created.Finish your profile by filling in the following fields
Forgot Password Earn Money,Free Notes
Password sent to your Email Id, Please Check your Mail
Updating Cart........ Please Wait........
Introduction to Solid State Physics
Introduction to Solid State Physics
Solid State Physics
Prof. Igor Shvets
Slide 1 Lecture 1Syllabus
Lattice structure: lattice with a basis.
Lattice structures of common chemical elements.
Concept of Bravais lattice, definition and examples.
Primitive vectors of Bravais lattice.
Primitive/Conventional unit cell.
WignerSeitz primitive cell.
Examples of common crystal structures.
Bodycentered cubic lattice.
Facecentered cubic lattice.
First Brillouin zone.
Lattice planes and Miller indices.
Slide 2 Lecture 1Syllabus
Determination of crystal structures by Xray diffraction.
Bragg formulation of Xray diffraction by a crystal.
Von Laue formulation of Xray diffraction by a crystal.
Equivalence of Bragg and Von Laue formulations.
Symmetry, elements of point groups
Geometrical structure factor.
Atomic form factor.
Stress and Strain.
Slide 3 Lecture 1Recommended Reading
Solid State Physics Ashcroft Mermin, HoltSaunders
• 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,
• Covers a huge amount in basic detail, good for problem solving.
Slide 4 Lecture 1Crystals
Solids can be categorised into either crystalline or non
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
Slide 5 Lecture 1Crystals
Despite an underlying crystalline
structure the crystal itself may
not appear regular in shape.
A closer look at the
substance reveals a
Slide 6 Lecture 1Examples
Principle component of
More on that later
Slide 7 Lecture 1Examples
Silicon Dioxide or Silica with a
definite crystalline structure.
Each Silicon atom is surrounded
by four Oxygen atoms.
Silicon Dioxide is an example of
Tetrahedral oxygen termination.
Oxidation State Electron
O 2 He.2s .2p
Si 4 Ne
Slide 8 Lecture 1Examples
Amorphous SiO (Silica gel)
As with the crystalline SiO
Most silicon atoms have 4
Most oxygen atoms have 2
Silica gel is an example of a non
The local symmetry is the same
as the crystalline SiO
However, translational symmetry
Slide 9 Lecture 1Oxygen Termination
The two forms of oxygen termination are Tetrahedral and
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 10 Lecture 1Examples
Aluminium Oxide (Al O )
Each Aluminium atom is
surrounded by six Oxygen
Aluminium Oxide is an
example of Octahedral oxygen
O 2 He.2s .2p
Al 3 Ne
Slide 11 Lecture 1Examples
Magnesium Oxide (MgO)
Structure is the same as
that of Sodium Chloride.
i.e. F.C.C. lattice with a two
Again more on that later
Oxidation State Electron Configuration
O 2 He.2s .2p
Mg 2 Ne
Slide 12 Lecture 1Examples
Magnetite ( Fe O )
32 oxygen anions form an F.C.C. lattice
This is an example
of Spinel structure.
8 tetrahedral interstices are occupied
by Fe ions
(Basic repeat unit)
16 octahedral interstices are occupied by
a = 0.8397nm
Fe and Fe ions in equal proportions
Slide 13 Lecture 1Spinel Group
Refers to a class of minerals with the general formulation
2+ 3+ 2
A B O
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 (Fe O ) the Iron is both A and B. That is,
A and B are the same metal under different charges.
3+ 3+ 2
Fe (Fe Fe )O
Slide 14 Lecture 1High 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
Assuming the system can be considered
a Carnot Engine and using current
average values of temperatures in the
T 2 system (T =35°C and T =540°C) the
efficiency is calculated to be 62.
Slide 15 Lecture 1High Performance Materials
As supplies of fossil fuels are
diminishing, there is large interest
in making the steam turbine a
more efficient process for
A way to increase this efficiency
is to operate the system with a
larger temperature gap, ie by
making T larger.
Slide 16 Lecture 1High Performance Materials
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
Slide 17 Lecture 1Lattice
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.
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.
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 18 Lecture 1Basis
You have a lattice
with more than one
atom/ion assigned to
each lattice point.
Atoms can be the
same or different.
will have larger/more
All crystal structure
consists of identical
Consider a point of the lattice
copies of the same
Then apply to the rest of the lattice
physical unit, called Now introduce a arbitrary basis of two atoms
the basis, assigned
to all the points of
Lattice + Basis = Crystal
Slide 19 Lecture 1Basis
While the basis is
assigned to each point
of the lattice there is
nothing to say that it is
anywhere near this
Consider the same
lattice and basis as
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 20 Lecture 1Basis
Even the most complex
crystal structures can be
broken down into a
lattice and a basis.
Lattice + Basis =
However, there is no
unique choice of Basis
Each of these Basis,
when combined with
this lattice, will yield
the same crystal.
Slide 21 Lecture 1Symmetry of a Basis
Consider a Basis that is itself symmetric and combine with the
lattice to create a simple crystal structure;
… they are also
the xaxis are
direction along the
direction (left or
yaxis (up or
Slide 22 Lecture 1Symmetry of a Basis
Now consider a basis that is not completely symmetric;
… However the
structure now has a
the xaxis are
than before and is
not the same from
direction (left or
the top and
Slide 23 Lecture 1Symmetry of a Basis
What if the basis is completely asymmetric
The properties of
the substance may
no longer be
direction they are
Slide 24 Lecture 1Bravais Lattice
The foundation of ANY Crystal structure is the Bravais lattice.
Definitions of Bravais Lattice
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 = n a + n a + n a
1 1 2 2 3 3
Where; a , a , a are three vectors not all in the same plane
1 2 3
and n , n , n are integer values.
1 2 3
3. A discrete set of vectors, not all in the same plane, closed
under vector addition or subtraction.
The vectors used to define a Bravais lattice,
a , a , a are called the primitive vectors.
1 2 3 ,
Slide 25 Lecture 1Bravais Lattice
Pictured: A general twodimensional
Bravais lattice of no particular symmetry.
The vectors a and a are primitive
One of the most common three
dimensional Cubic Bravais
lattices, the Simple Cubic lattice.
All a , a , a are of equal length
1 2 3 ,
Slide 26 Lecture 1NonBravais 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 R
In 3D an example of a non
Bravais lattice is the Hexagonal
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 27 Lecture 12D NonBravais Lattice with a Basis
All NonBravais lattices
can be created from a
Bravais lattice with a
The red points form a
simple 2D bravais lattice.
Adding a twopoint basis
to each lattice point…
and it starts to look
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 twopoint basis.
Slide 28 Lecture 12D NonBravais Lattice with a Basis
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 superimpose the
picture of the honeycomb
over ours it is obvious
that they do not match.
Slide 29 Lecture 1Exercise
In each of the following cases indicate whether the structure is a
a) Basecentered Cubic (Simple cubic with additional points in the
centres of the horizontal faces 2)
b) Sidecentered Cubic (Simple cubic with additional points in the
centres of the vertical faces 4)
c) Edgecentered Cubic (Simple cubic with additional points at the
midpoints of the lines joining the simple cubic points 12)
If it is a Bravais lattice give three primitive vectors.
If not describe it as Bravais with the smallest possible basis.
Slide 30 Lecture 1Questions/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
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
Slide 31 Lecture 1