Crystal Structures

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APPLIED PHYSICS

CODE : 07A1BS05

I B.TECH

CSE, IT, ECE & EEE

UNIT-1: CHAPTER 2.1

NO. OF SLIDES :33

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S.No. Module Lecture

No.

PPT Slide No.

1 Introduction-space lattice –unit cell

L5 3-10

6 Lattice parameters. bravais lattices

L6 11-27

7 Structure and packing fractions.

L7 28-30

8. Miller indices. L8-9 31-33

UNIT INDEXUNIT INDEXUNIT-I UNIT-I

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INTRODUCTION

Matter is classified into three kinds, they are solids, liquids and gases. In solids, all the atoms or molecules are arranged in a fixed manner. Solids have definite shape and size, where as in liquid and gasses atoms or molecules are not fixed and cannot form any shape and size.

On basis of arrangement of atoms or molecules, solids are classified into two categories, they are crystalline solids and amorphous solids.

Lecture-5

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CRYSTALLINE SOLIDS AMORPHOUS SOLIDS

1. In crystalline solids, the atoms or molecules are arranged in a regular and orderly manner in 3-D pattern, called lattice.

2. These solids passed internal spatial symmetry of atomic or molecular orientation.

3. If a crystal breaks, the broken pieces also have regular shape.

Eg: M.C : Au, Ag,Al,

N.M.C: Si, Nacl, Dia.

1. In amorphous solids, the atoms or molecules are arranged in an irregular manner, otherwise there is no lattice structure.

2. These solids do not posses any internal spatial symmetry.

3. If an amorphous solid breaks, the broken pieces are irregular in shape.

Eg : Glass, Plastic, Rubber.

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LATTICE POINTS :

Lattice points denote the position of atoms or molecules in the crystals.

SPACE LATTICE :

The angular arrangement of the space positions of the atoms or molecules in a crystals is called space lattice or lattice array.

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2D-SPACE LATTICE :

It is defined as an infinite array of points in 2-D space in which every point has the same environment w.r.t. all other points.

The dots represent the lattice points in which atoms can be accommodated. Taking O as an arbitrary origin in XY – plane constructed.

The two translations vectors ā and ē are taken along X-axis and Y-axis respectively. The resultant vector T can be represented as

T=n1ā +n2 Where n1, n2 are arbitrary integers.

OP b

b

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3D- Space Lattice

It is defined as an infinite array of points in 3D-Space in which every point has the same environment w.r.t. all other points.

In this case the resultant vector can be expressed as T=n1ā +n2 +n3 . Where n1, n2, n3 are arbitrary integers and, ā, & are translational vector along X,Y,Z-axis respectively

b ccb,

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BASIS : • Certain atoms or molecules are attached

to each lattice point in the crystal structure. These atoms or molecules attached to any lattice point form the basis of a crystal lattice. Hence, crystal structure = Lattice + Basis.

• In order to convert the geometrical array of points molecules are located on the lattice points.

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• The repeating unit assembly – atom, molecule, ion or radical – that is located at each lattice point is called the BASIS.

• The basis is an assembly of atoms identical in composition, arrangement and orientation. Thus, Again we say that the crystal structure is formed by logical relation

Space lattice + Basis = CRYSTAL STRUCTURE.

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Unit Cell :

Unit cell of a crystal is the smallest volume of a crystalline solid or geometric figure from which the entire crystal is built up by translational repetition in three dimensions.

• Since the unit cell which reflects the structure of

the crystal structure of the crystal lattice has all the structural properties of the given crystal lattice, it is enough to study the shape and properties of the unit cell to get the idea about the whole crystal

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LATTICE PARAMETERS OF AN UNIT CELL

The lines drawn parallel to the lines of intersection of any three faces of the unit cell which do not lie in the same plane are called crystallographic axes.

An arbitrary arrangement of crystallographic axes marked X,Y,&Z. The angles between the three crystallographic axes are known as interfacial angles or interaxial angles.

Lecture-6

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• The angle between the axes Y and Z = α

• The angle between the axes Z and X = β

• The angle between the axes X and Y = γ The intercepts a,b&c define the dimensions of

an unit cell and are known as its primitive or characteristic intercepts on the axes. The three quantities a,b&c are also called the fundamental translational vectors.

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BRAVAIS LATTICES

• A 3dimensional lattice is generated by repeated translation of three non-coplanar vectors a,b &c.

• There are only 14 distinguishable ways of arranging points in 3d space.

• These 14 space lattices are known as Bravais lattices.

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SIMPLE CUBIC

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BODY CENTRED CUBIC

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FACE CENTRED CUBIC

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TETRAGONAL

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BODY CENTRED TETRAGONAL

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ORTHORHOMBIC

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BODY CENTRED ORTHORHOMBIC

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BASE CENTRED ORTHORHOMBIC

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FACE CEN TRED ORTHORHOMBIC

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MONOCLINIC

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BASE CENTRED MONOCLINIC

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TRICLINIC

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RHOMBOHEDRAL

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HEXAGONAL

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Lecture-7

Atomic packing factor is the ratio of volume occupied by the atoms in an unit cell to the total volume of the unit cell. It is also called packing fraction.

• The arrangement of atoms in different layers and the way of stacking of different layers result in different crystal manner.

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• Metallic crystals have closest packing in two forms (i) hexagonal close packed and

• (ii) face- centred cubic with packing factor 74%.

• The packing factor of simple cubic structure is 52%.

• The packing factor of body centred cubic structure is 68%.

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MILLER INDICES

In a crystal orientation of planes or faces can be described interms of their intercepts on the three crystallographic axes.

Miller suggested a method of indicating the orientation of a plane by reducing the reciprocal of the intercepts into smallest whole numbers.

o These indices are called Miller indeces generally represented by (h k l).

Lecture-8

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• All equally spaced parallel planes have the same miller indices.

• . If a normal is drawn to a plane (h k l), the direction of the normal is

[h k l].

• Separation between adjacent lattice planes in a cubic crystal is given by d= u/ ---h2+k2+l2. where a is the lattice constant and (h k l) are the Miller indices.

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Important features in miller indices

1. When a plane is parallel to any axis, the intercept of the plane on that axis is infinity. Hence its Miller index for that axis is zero.

2. When the intercept of a plane on any axis is negative a bar is put on the corresponding Miller index.

3. All equally spaced parallel planes have the same index number (h k l).

Lecture-9

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4. If a plane passes thought origin, it is defined in terms of a parallel plane having non-zero intercept.

5. If a normal is drawn to plane (h k l), the direction of the normal is (h k l).