lecture+1+mak crystal+structure

7/30/2019 Lecture+1+MAK Crystal+Structure

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PHY 3201 FIZIK KEADAAN PEPEJAL

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
http://www.alaskanessences.com/gembig/Pyrite.jpg


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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


7/34PHY 3201 FIZIK KEADAAN PEPEJAL

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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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?



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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

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