Lecture+1+MAK Crystal+Structure
Transcript of Lecture+1+MAK Crystal+Structure
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CHAPTER 1: CRYSTAL
STRUCTURE
The majority of commonly used materials are in the solidstate. Materials may be broadly classified as:
Metals and alloys Ceramics, and Polymers
Depending on the regularity with which the atoms ormolecules are arranged in solids, they may be broadly
classified as: Crystalline (crystal or polycrystalline form) Non-crystalline (amorphous)
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SOLID MATERIALS
CRYSTALLINE POLYCRYSTALLINEAMORPHOUS(Non-crystalline)
Single Crystal
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CHAPTER 1: CRYSTAL
STRUCTUREWhy crystalline solids?
The important electronic properties of solids are bestexpressed in crystals. Thus the properties of the mostimportant semiconductors depend on the crystalline
structure of the host, essentially because electronshave short wavelength components that responddramatically to the regular periodic atomic order of thespecimen.
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Crystalline material is one in which the atoms are situated
in a repeating or periodic array over large atomic
distances.
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Single crystal: the regular periodic arrangement of atoms
extends over the entire volume of solids i.e. it possesseslong range order.
Polycrystalline solid: made up of an aggregate of a large
number of tiny single crystals oriented in different
directions and separated by well-defined boundaries
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Amorphous materials: have no regular arrangement of
atoms or molecules. However, the arrangement is not
completely disordered; i.e. short-ranged order of about 1-
1.5 nm
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Lattice Translation
Vectors
The geometrical arrangement of atoms inan ideal crystal (one that extends to
infinity) can be described using 2
elementsa lattice and a basis.
Crystal Structure = Lattice + Basis
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Crystal Lattice
An infinite array ofpoints in space,
Each point has identicalsurroundings to all
others.
Arrays are arrangedexactly in a periodic
manner.
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Basis
The atom or group of atoms or molecules
attached to each lattice point in ordergenerate the crystal structure.
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Lattice Translation
Vectors
+ =
Crystal StructureBasisLattice
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Lattice Translation
VectorsThe lattice is defined by three fundamental translation
vectors
321 and, aaa
If defines the vector ending on one lattice point,then the points
332211' auauaurr
defines all the points on the lattice, where the ui areintegers. A key feature is that all the points are
identical i.e. the lattice appears identical when views
from the point or as when viewed from the point
(1)
r
'r
r
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a1
a2
The set of points defined by (1) for all u1, u2, u3defines the lattice
If an observer at
any point A is
translated to any
point B, he will not
be able to detect
any change in the
environment i.e.
the environment ofall the points in a
given lattice are
identical.
A
B
2113' aarr
2135' aar
r
'r
'r
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THE BASIS
For one basis, there is at least one atom but forcomplicated crystal e.g. organic proteins, there can bethousands of atoms per basis.
The position of the centre of an atom j of the basisrelative to the associated lattice point is
321azayaxr jjjj
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Primitive Lattice Cell
The paralellepiped defined by
primitive axes , and is called
a primitive cell or a unit cell.
The unit cell is a fundamental block
which, when repeated in the threedirections, will generate the entire
lattice.
A primitive unit cell is a minimum
volume cell such that there is nocell of smaller volume that can be
used as a building block for crystal
structures
1a
2a
3a
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Primitive Lattice Cell
There are many ways of choosing the primitiveaxes and primitive cell for a given lattice.
The number of atom in a primitive cell orprimitive basis is always the same for a givencrystal structure.
There is always one lattice point per primitivecell
Cell volume is given as 321 aaaVc
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A, B, and C are primitive
unit cells.The volumes of A, B, and C
are the same.
The choice of origin is
different, but it doesntmatter.There is only one lattice
point in the primitive unit
cells.
D, E, and F are not. Why?
Primitive Lattice Cell
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Wigner-Seitz cellsA simple way to find the
primitive cell which is calledWigner-Seitz cell can be doneas follows;
1. Choose a lattice point.
2. Draw lines to connect theselattice point to its neighbours.
3. At the mid-point and normalto these lines draw newlines.
4. The volume enclosed iscalled as a Wigner-Seitzcell.
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Symmetry in Crystals
There are several ways in which an object may be
repeated in space. These are called symmetry
operations.
Symmetry operation Symmetry elementTranslation Displacement
Rotation Rotation axis
reflection Mirror plane
Inversion Inversion point (or
symmetry center)
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S
S
S
S
S
S
S
S
S
S
S
S
S
S
Translation
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This is an axis such that, if the cell is rotated around itthrough some angles, the cell remains invariant.
The axis is called n-fold if the angle of rotation is 2/n.
Rotation
90
120 180
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Axis of rotation
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Axis of rotation
How about 5-fold symmetry?
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5-fold symmetry
Empty space
not allowed
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Reflection
A plane in a cell such that, when a mirror reflection inthis plane is performed, the cell remains invariant.
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Inversion Center
A center of symmetry: A point at the center of the molecule.(x,y,z) --> (-x,-y,-z)
Center of inversion can only be in a molecule. It is notnecessary to have an atom in the center (benzene, ethane).
Tetrahedral, triangles, pentagons don't have a center ofinversion symmetry. All Bravais lattices are inversionsymmetric.
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Mathematicians discovered that you can only fill space by
using rotations of unit cells by 2, 2/2, 2/3, 2/4, and2/6 radians (or, by 360o, 180o, 120o, 90o, and 600)
But, rotations of the kind 2/5 or 2/7 do not fill space!
In 2D, there are only 5 distinct lattices. These are defined
by how you can rotate the cell contents (and get the
same cell back), and if there are any mirror planes within
the cell.From now on, we will call these distinct lattice types
Bravais lattices.
Unit cells made of these 5 types in 2D can fill space. All
other ones cannot.
Two-Dimensional
Lattice Types
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1a
2a
oblique lattice
a1 a2, =arbitrary
Two-Dimensional
Lattice Types
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Two-Dimensional
Lattice Types
1a
1a
2a
2a
Square lattice
a1 = a2, =90o
Hexagonal lattice
a1 =a2, =120o
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Two-Dimensional
Lattice Types
2a
1a
1a
1a
2a
2a
Rectangular lattice
a1 a2, =90o Centered rectangular latticea1 a2, =90o
Oblique
a1 a2
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Summary of
Lecture 1Crystalline Atoms are arranged in a periodic array Crystals consists of a infinite repetition of
individual structural units arranged periodically
in space All crystal structures can be described in terms
of a lattice, with an identical group of atoms (ormolecules) attached to every lattice point called
the basis The lattice is formally known as a BravaisLattice
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Summary of
Lecture 1
A key feature of a Bravais lattice is that allpoints are identical
Lattice + basis = crystal crystal structure Primitive Lattice Cell the parallelpiped difined
by the primitive translation vectors defines aprimitive cell or primitive unit cell. It is aminimum volume cell. It contains one latticepoint per primitive cell
Conventional Unit Cell a non-primitive cellthat displays the symmetry of the Bravais latticeand fills all space when translated by a subsetof Bravais lattice translation vectors , T
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3D-Unit cell