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Representator:MD. MOHYMENUL ISLAM

ID: PH 120043rd YEAR 1st SEMESTERDEPT. OF PHYSICSMBSTU

CrystalTranslational VectorCrystal StructureCrystal LatticeUnit cellLattice ConstantSymmetry OperationPacking FactorMiller IndicesInter-planar Spacing

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3CRYSTAL SYSTEM

SOLID MATERIALS

CRYSTALLINE

Single Crystal

POLYCRYSTALLINEAMORPHOUS

(Non-crystalline)

A crystal is a solid in which atoms are arranged in some regular repetition pattern in all directions.

Crystal

Crystals (at a given temperature, pressure, and volume) have stronger bonds between molecules and atoms than liquids. It’s require more energy to break the bonds.

Crystalline material is a material comprised of one or many crystals. Ex:Diamond,quartz,mica etc.

Amorphous have order only within a few atomic or molecular dimensions. Ex: Glass,plastics,rubbers.

Polycrystallne is a material made up of an aggregate of many small single crystals (also called crystallites or grains).Ex:Metals and ceramics.

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*Crystal Types

Single Pyrite Crystal

AmorphousSolid

Polycrystalline Pyrite form

(Grain)

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* Difference between crystalline & non-crystalline

1.Long range order.

2.Fixed melting point.

3.Atoms or molecules are periodically arranged.

1.Short range order.

2.No fixed melting point.

3.Atoms or molecules are randomly arranged.

Crystalline Non-crystalline

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Translational Lattice Vectors

A lattice translation operation is defined by the displacement of a crystal by a crystal translation vector.

Rn = n1 a + n2 b

This is translational symmetry.

The vectors a, b are known as lattice vectors and (n1, n2) is a pair of integers whose values depend on the lattice point.

P

Point D(n1, n2) = (0,2)

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*Crystal Structure*A CRYSTAL STRUCTURE is a periodic arrangement of atoms in the crystal that can be described by a LATTICE

Lattice: A 3D translationally periodic arrangement of points in space.Basis: A group of atoms associated with each lattice point to represent crystal structure.

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Crystal Lattice

Bravais Lattice (BL) All atoms are of the same kind All lattice points are equivalent

Non-Bravais Lattice (non-BL)

Atoms can be of different kind Some lattice points are not equivalentA combination of two or more BL

Bravais LatticeNon-Bravais Lattice

UNIT CELL

The smallest component of the crystal which when stacked together with pure translational repetition reproduces the whole crystal.

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S

a

b

S

S

S

S

S

S

S

S

S

S

S

S

S

S

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THREE COMMON UNİT CELL İN 3D • Primitive (P) unit cells contain only a single lattice point.• Internal (I) unit cell contains an atom in the body center.• Face (F) unit cell contains atoms in the all faces of the planes composing the cell.

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Crystal Structure 11

Unit cell exist in only seven shapes

The numbers a,b,c specifying the size of a unit cell (in fact, conventional unit cell) are called its lattice constant

For cubic lattice, the lattice constant,

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Where, ρ=density of the latticen= number of particles M= molecular weight of the crystal NA =Avogadro number

a=(nM/NAρ)1/3

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Symmetry Operation: A symmetry operations is one which leaves the crystal unchanged such as translation, rotation, reflection or inversion.

5-fold symmetry:

Symmetry Operations

N-fold axes with n=5 or n>6 does not occur in crystals

Adjacent spaces must be completely filled (no gaps, no overlaps).

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*Atomic Packing Factor*Atomic Packing Factor (APF) is defined as the volume of atoms in the unit cell divided by the volume of the unit cell.

Simple Cubic Structure (SC)Close-packed directions are cube edges.

• Coordination = 6 (nearest neighbors)

a = 2r, r = a/2Atomic Radius, r=0.5a

a

Atomic Packing Factor (APF):SC

• APF = 0.52, That means the percentage of packing is 52%

APF = a3

4

3 p (0.5a) 31

atoms

unit cellatom

volume

unit cell

volume

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

close-packed directions

a

R=0.5a

Number of lattice point 8 x 1/8 = 1 atom/unit cell

Body Centered Cubic Structure (BCC)

• Coordination # = 8

• Atoms touch each other along cube diagonals.

Ex: Cr, Fe , Molybdenum

2 atoms/unit cell: 1 center + 8 corners x 1/8

Atomic Radius ,r =a x (3)1/2/4

Atomic Packing Factor: BCC

a

APF =

4

3p ( 3a/4)32

atoms

unit cell atom

volume

a3unit cell

volume

length = 4R =Close-packed directions:

3 aaR

a 2

a 3

• APF for a body-centered cubic structure = 0.68

Face Centered Cubic Structure (FCC)

• Coordination = 12

• Atoms touch each other along face diagonals.

Ex: Al, Cu, Ni,Ag

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

Atomic Radius ,r = a x (2)1/2/4

Atomic Packing Factor: FCC

APF =

4

3 p ( 2a/4)34

atoms

unit cell atom

volume

a3unit cell

volume

Close-packed directions: length = 4R = 2 a

Number of lattice point : 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

a

2 a

• APF for a face-centered cubic structure = 0.74

Miller Indices• Miller Indices is a group of smallest integers which represent a direction or a plane.

To determine Miller indices of a plane, take the following steps:

1) Determine the intercepts of the plane along each of the three crystallographic directions2) Take the reciprocals of the intercepts3) If fractions result, multiply each by the denominator of the smallest fraction.

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(233)

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Inter-planar Spacing

For orthorhombic, tetragonal and cubic unit cells (the axes are all mutually perpendicular), the inter-planar spacing is given by:

h, k, l = Miller indices

a, b, c = unit cell dimensions

222 lkh

adhkl

• For cube a = b = c than

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REFERENCES

Sourcewww.Google.comhttp://en.wikipedia.orgSolid State Physics & Electronics

By R.K. PURI” Solid State Physics ” By R.L.

SINGHALSeventh Revised & Enlarged Edition-2003

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