CONDENSED MATTER CONDENSED MATTER PHYSICSPHYSICS

PHYSICS PAPER A

BSc. (III) (NM and CSc.)

Harvinder Kaur

Associate Professor in Physics

PG.Govt College for Girls

Sector -11, Chandigarh

Prof. Harvinder Kaur

PG.Govt College for Girls

Sector -11, Chandigarh

OUTLINEOUTLINE

Crystal StructureCrystal Structure

Unit CellUnit Cell

Symmetry Operations Symmetry Operations

Bravais LatticeBravais Lattice

Characteristics of Unit Cell of cubic systemCharacteristics of Unit Cell of cubic system

Closed packed structureClosed packed structure

Miller IndicesMiller Indices

CRYSTAL STRUCTURECRYSTAL STRUCTURE

Crystal structure Crystal structure is a unique arrangement of atoms, molecules or ions is a unique arrangement of atoms, molecules or ions constructed by the infinite repetition of identical structural units(called constructed by the infinite repetition of identical structural units(called unit cell) in space .The structure of all crystals can be described in terms unit cell) in space .The structure of all crystals can be described in terms of of latticelattice & & basisbasis..

lattice lattice : regular periodic arrangements of identical points in space: regular periodic arrangements of identical points in space

Basis Basis : A group of atoms or ions: A group of atoms or ions

UNIT CELLUNIT CELL

Primitive Unit cell has one lattice pointPrimitive Unit cell has one lattice point

A A primitive unit cell primitive unit cell of a particular crystal of a particular crystal structure is the smallest possible volume structure is the smallest possible volume one can construct with the arrangement of one can construct with the arrangement of atoms in the crystal such that, when atoms in the crystal such that, when stacked, completely fills the space. This stacked, completely fills the space. This primitive unit cell will not always display all primitive unit cell will not always display all the symmetries inherent in the crystal. the symmetries inherent in the crystal.

Unit cell : Unit cell : A building block that can be A building block that can be periodically duplicated to result in the periodically duplicated to result in the crystal structure, is known as the unit cell.crystal structure, is known as the unit cell.

Unit Cell is of two typesUnit Cell is of two types

Primitive Unit CellPrimitive Unit Cell

Non-Primitive Unit CellNon-Primitive Unit Cell

Non Primitive Unit cell has more than one Non Primitive Unit cell has more than one lattice pointlattice point

A Physicist Wigner Seitz gave a geometrical A Physicist Wigner Seitz gave a geometrical way to design a primitive unit cell known as way to design a primitive unit cell known as Wigner Seitz cell Wigner Seitz cell

Steps for the construction of Wigner Seitz Steps for the construction of Wigner Seitz CellCell

Draw lines to connect a given lattice point Draw lines to connect a given lattice point to all nearby lattice pointsto all nearby lattice points

At the midpoint and normal to these lines At the midpoint and normal to these lines draw new lines or planesdraw new lines or planes

The smallest volume enclosed in this way The smallest volume enclosed in this way is is Wigner- Seitz primitive cellWigner- Seitz primitive cell

WIGNER SEITZ PRIMITIVE CELLWIGNER SEITZ PRIMITIVE CELL

SYMMETRY OPERATIONSSYMMETRY OPERATIONS

A symmetry operation is the one that leaves the crystal and its environment invariant. Symmetry operations performed about a point are called point group symmetry operations like Rotation, Reflection and Inversion

Types of Symmetry operations

Translation Symmetry

Rotation

Reflection

Inversion

TRANSLATION SYMMETRYTRANSLATION SYMMETRY

The translation symmetry is the manifestation of the order

of crystalline solids.

r’= r + T= r + n1a +n2b +n3c

b

a

c

Translational operator, T is defined in terms of three fundamental vectors, a,b and c

T = n1 a+n2 b+n3 c

Translational symmetry means that when the operator T is applied on any point r in the crystal, the resulting point r’ is exactly identical in all respects to the original point r

ROTATIONROTATION

A lattice is said to possess the rotational symmetry about an axis if the rotation of the lattice by some angle leaves it invariant. Since the lattice remains invariant by rotation of 2, so must be equal to 2/n with n an integer. The integer n is called the multiplicity of the rotation axis.

REFLECTIONREFLECTION

A lattice is said to possess reflection symmetry about a plane (or a line in two dimensions) if it is left unchanged after being reflected in a plane. In other words the plane divides the lattice into two identical halves which are mirror images of each other.

INVERSIONINVERSION

A crystal structure possesses an inversion symmetry if for each point located at r relative to a lattice point there exists an identical point at –r. Inversion is applicable in three dimensional lattices only.

BRAVAIS LATTICEBRAVAIS LATTICE

Bravais lattices :Bravais lattices :The space lattices which are invariant under one The space lattices which are invariant under one or more point of the symmetry operation are known as Bravais or more point of the symmetry operation are known as Bravais lattices. There are five Bravais lattice in two dimensions and 14 lattices. There are five Bravais lattice in two dimensions and 14 unique Bravais lattices in three dimensions unique Bravais lattices in three dimensions

In two dimensions, there are five Bravais lattices. These are 1. Oblique 2. Rectangular 3.Centered Rectangular 4. Hexagonal 5.Square

CRYSTAL SYSTEMCRYSTAL SYSTEM

In three dimensions the 14 Bravais lattices are grouped into 7 In three dimensions the 14 Bravais lattices are grouped into 7 crystal systems according to the seven types of conventional crystal systems according to the seven types of conventional cells. They are :cells. They are :

Triclinic - 1 Bravais Lattice, least symmetricTriclinic - 1 Bravais Lattice, least symmetric

Monoclinic – 2 Bravais LatticesMonoclinic – 2 Bravais Lattices

Orthorhombic – 4 Bravais LatticesOrthorhombic – 4 Bravais Lattices

Rhombohedral/Trigonal -1 Bravais LatticeRhombohedral/Trigonal -1 Bravais Lattice

Tetragonal – 2 Bravais LatticesTetragonal – 2 Bravais Lattices

Hexagonal – 1 Bravais LatticesHexagonal – 1 Bravais Lattices

Cubic - 3 Bravais Lattices, most symmetricCubic - 3 Bravais Lattices, most symmetric

CRYSTAL SYSTEM CONTINUEDCRYSTAL SYSTEM CONTINUED

SimpleSimple Base-CentereBase-Centeredd Base-Base-

CenteredCenteredFace-CenteredFace-Centered

CRYSTAL SYSTEM CONTINUEDCRYSTAL SYSTEM CONTINUEDRhombohedral Tetragonal

TRICLINICTRICLINIC

a b c

MONOCLINICMONOCLINIC

SimpleSimple Base CenteredBase Centered

a b c

ORTHORHOMBICORTHORHOMBIC

SimpleSimple Base-Base-CenteredCentered

Body-Body-CenteredCentered

Face-Face-CenteredCentered

= β= = 90

RHOMBOHEDRAL or TRIGONALRHOMBOHEDRAL or TRIGONAL

TETRAGONALTETRAGONAL

SimpleSimple Body-CenteredBody-Centered

= β= = 90

HEXAGONALHEXAGONAL

= β=90, = 120

CUBICCUBIC

CHARACTERISTICS OF THE UNIT CHARACTERISTICS OF THE UNIT CELL OF THE CUBIC SYSTEMCELL OF THE CUBIC SYSTEM

Volume : The volume of unit cell is a3

Atoms per unit cell : Simple Cubic - 1

Body Centered Cubic – 2

Face Centered Cubic - 4

Cooridination Number : It is equal to the number of nearest neighbour that surrounds each atom.

Simple Cubic - 6

Body Centered Cubic – 8

Face Centered Cubic - 12

FCC

Atomic Radius (r) :

Simple Cubic - r= a/2

Body Centered Cubic – r = (3/4)a

Face Centered Cubic - r = (2/4)a

Atomic packing factor = Volume of atoms in a unit cell

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Volume of the unit cell

For Simple cubic

P.F = (1x(4/3)r3 )/a3 = /6 = 0.524

For Body centered cubic

P.F = (2x(4/3)r3 )/a3 = 3/8 = 0.680

For Face centered cubic

P.F = (4x(4/3)r3 )/a3 = (2)/6 = 0.740

CLOSE PACKED STRUCTURECLOSE PACKED STRUCTURE

CLOSE PACKED STRUCTURE CLOSE PACKED STRUCTURE ABAB STACKING GIVE RISE TO HEXAGONAL CLOSED ABAB STACKING GIVE RISE TO HEXAGONAL CLOSED

PACKED STRUCTUREPACKED STRUCTURE

CLOSE PACKED STRUCTURE CLOSE PACKED STRUCTURE ABCABC.. STACKING GIVE RISE TO FACE CENTERED ABCABC.. STACKING GIVE RISE TO FACE CENTERED

CUBIC STRUCTURE CUBIC STRUCTURE

NaCl Crystal StructureNaCl Crystal Structure

The NaCl lattice is face –centered cubic; the basis consists of one Na atom and in Cl atom separated by one-half the body diagonal of unit cube. There are four units of NaCl in each unit of cube, with atoms in the positions

Cl: 000 ½½0 ½0½ 0½½

Na : ½½½ 00½ 0½0 ½00

The NaCl structure has ionic bonding with each atom having 6 nearest neighbour and 12 next nearest neighbour. It has primitive unit cell which is simple cubic

Atomic Packing fraction = 52.4%

Diamond Crystal StructureDiamond Crystal Structure

The Diamond lattice is face –centered cubic; the basis consists of two identical C atoms separated by one-fourth the body diagonal of unit cube.

C: 000 ½½0 ½0½ 0½½

C : ¼¼¼ ¾¾¼ ¾¼¾ ¼¾¾

The Diamond structure has tetrahedral bonding with each atom having 4 nearest neighbour and 12 next nearest neighbour. Atomic packing fraction = 34%

Miller indices are a notation system in crystallography for planes in crystal (Bravais) lattices.

MILLER INDICESMILLER INDICES

Steps for calculating Miller Index

Take any lattice point as origin in the crystal lattice and erect coordinate axis from this point in the direction of three basis vectors, a,b and c

Identify the intercepts on these axis made by a plane of the set of a parallel planes of interest in terms of lattice constant

Take the reciprocals of these intercepts and reduce these into smallest set of integers h,k,l

The miller Indices of a set of parallel planes – (h k l)

MILLER INDICESMILLER INDICES

Planes with different Miller indices in cubic

crystals

EXAMPLES

EXAMPLES

Examples

LATTICE DIRECTION Generally the square brackets are used to indicate the direction i.e., [h,k,l]