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## INTRODUCTION TO CRYSTAL PHYSICS

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**INTRODUCTION TO CRYSTAL PHYSICS**What is Crystal Physics? ‘Crystal Physics’ or ‘Crystallography’ is a branch of physics that deals with the study of all possible types of crystals and the physical properties of crystalline solids by the determination of their actual structure by using X-rays, neutron beams and electron beams. PH 0101 UNIT 4 LECTURE 1**CLASSIFICATION OF SOLIDS**• Solids can broadly be classified into two types based on the • arrangement of units of matter. • The units of matter may be atoms, molecules or ions. • They are, • Crystalline solids and • Non-crystalline (or) Amorphous solids PH 0101 UNIT 4 LECTURE 1**CRYSTALLINE SOLIDS**• A substance is said to be crystalline when the • arrangement of units of matter is regular and • periodic. • A crystalline material has directional properties and • therefore called as anisotropic substance. • A crystal has a sharp melting point. • It possesses a regular shape and if it is broken, all • broken pieces have the same regular shape. PH 0101 UNIT 4 LECTURE 1**CRYSTALLINE SOLIDS**• A crystalline material can either be a single • (mono) crystal or a polycrystal. • A single crystal consists of only one crystal, • whereas the polycrystalline material consists of • many crystals separated by well-defined • boundaries. • Examples • Metallic crystals – Cu, Ag, Al, Mg etc, • Non-metallic crystals – Carbon,Silicon,Germanium, PH 0101 UNIT 4 LECTURE 1**NON CRYSTALLINE SOLIDS**• In amorphous solids, the constituent particles • are not arranged in an orderly manner. They • are randomly distributed. • They do not have directional properties and so • they are called as `isotropic’ substances. • They have wide range of melting point and do • not possess a regular shape. • Examples: • Glass, Plastics, Rubber etc., PH 0101 UNIT 4 LECTURE 1**ATOMIC ARRANGEMENT IN CRYSTALS**• mono (or) single crystals • (b) polycrystalline solids • (c) amorphous solids PH 0101 UNIT 4 LECTURE 1**CRYSTALS**• It is a substance in which the constituent particles are • arranged in a systematic geometrical pattern. PH 0101 UNIT 4 LECTURE 1**=**+ CRYSTAL STRUCTURE Crystal structure = Lattice + Basis PH 0101 UNIT 4 LECTURE 1**UNIT CELL**• A unit cell is defined as a fundamental building block • of a crystal structure, which can generate the • complete crystal by repeating its own dimensions in • various directions. PH 0101 UNIT 4 LECTURE 1**UNIT CELL**PH 0101 UNIT 4 LECTURE 1**CRYSTALLOGRAPHIC AXES**• Consider a unit cell consisting of three mutually • perpendicular edges OA, OB and OC as shown in • figure. • Draw parallel lines along the three edges. • These lines are taken as crystallographic axes and they • are denoted as X, Y and Z axes. PH 0101 UNIT 4 LECTURE 1**LATTICE PARAMETERS**• Consider the unit cell as shown in figure. • Let OA, OB and OC are the intercepts made by the • unit cell along X, Y and Z axes respectively. • These intercepts are known as primitives. In • crystallography the intercepts OA, OB and OC are • represented as a , b and c . PH 0101 UNIT 4 LECTURE 1**LATTICE PARAMETERS**• The angle between X and Y axes is represented as . • Similarly the angles between Y and Z and Z and X axes • are denoted by and respectively as shown in the • above figure. These angles , and are called as • interaxial angles orinterfacial angles. • To represent a lattice, the three interfacial angles and • their corresponding intercepts are essential. These six • parameters are said to be lattice parameters. PH 0101 UNIT 4 LECTURE 1**PRIMITIVE CELL**• It is the smallest unit cell in volume constructed • by primitives. It consists of only one full atom PH 0101 UNIT 4 LECTURE 1**PRIMITIVE CELL**• A primitive cell is one, which has got the points or • atoms only at the corners of the unit cell. • If a unit cell consists of more than one atom, then it is • not a primitive cell. • Example for primitive cell :Simple Cubic unit cell. • Examples for non-primitive cell:BCC and FCC unit cell. PH 0101 UNIT 4 LECTURE 1**CRYSTALS SYSTEMS**• The seven systems are, • Cubic (isometric) • Tetragonal • Orthorhombic • Trigonal (rhombohedral) • Hexagonal • Monoclinic and • Triclinic PH 0101 UNIT 4 LECTURE 1**CRYSTALS SYSTEMS**• The space lattices formed by unit cells are marked by the following symbols. • Primitive lattice:Phaving lattice points only at the • corners of the unit cell. • Body centred lattice:I having lattice points at • the corners as well as at the body centre of the unit • cell. PH 0101 UNIT 4 LECTURE 1**CRYSTALS SYSTEMS**• Face centred lattice:F having lattice points at the • corners as well as at the face centres of the unit cell. • Base centred lattice:C having lattice points at the • corners as well as at the top and bottom base • centres of the unit cell. PH 0101 UNIT 4 LECTURE 1**BRAVAIS LATTICES**• Bravais in 1948 showed that 14 types of unit cells • under seven crystal systems are possible. They • are commonly called as `Bravais lattices’. PH 0101 UNIT 4 LECTURE 1**MILLER INDICES**• Miller indices is defined as the reciprocals of • the intercepts made by the plane on the three • axes. PH 0101 UNIT 4 LECTURE 2**MILLER INDICES**• Procedure for finding Miller Indices • Step 1: Determine the interceptsof the plane • along the axes X,Y and Z in terms of • the lattice constants a,b and c. • Step 2: Determine the reciprocals of these • numbers. PH 0101 UNIT 4 LECTURE 2**MILLER INDICES**• Step 3:Find the least common denominator (lcd) • and multiply each by this lcd. • Step 4:The result is written in paranthesis.This is • called the `Miller Indices’ of the plane in • the form (h k l). • This is called the `Miller Indices’ of the plane in the form • (h k l). PH 0101 UNIT 4 LECTURE 2**EXAMPLE**( 1 0 0 ) plane Plane parallel to Y and Z axes PH 0101 UNIT 4 LECTURE 2**MILLER INDICES**• IMPORTANT FEATURES OF MILLER INDICES • A plane passing through the origin is defined in terms of a • parallel plane having non zero intercepts. • All equally spaced parallel planes have same ‘Miller • indices’ i.e. The Miller indices do not only define a particular • plane but also a set of parallel planes. Thus the planes • whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3 etc., are all • represented by the same set of Miller indices. PH 0101 UNIT 4 LECTURE 2**MILLER INDICES**• IMPORTANT FEATURES OF MILLER INDICES • It is only the ratio of the indices which is important in this • notation. The (6 2 2) planes are the same as (3 1 1) planes. • If a plane cuts an axis on the negative side of the origin, • corresponding index is negative. It is represented by a bar, • like (1 0 0). i.e. Miller indices (1 0 0) indicates that the • plane has an intercept in the –ve X –axis. PH 0101 UNIT 4 LECTURE 2**MILLER INDICES OF SOME IMPORTANT PLANES**PH 0101 UNIT 4 LECTURE 2**CRYSTAL DIRECTIONS**• In crystal analysis, it is essential to indicate certain • directions inside the crystal. • A direction, in general may be represented in terms of • three axes with reference to the origin.In crystal system, • the line joining the origin and a lattice point represents • the direction of the lattice point. PH 0101 UNIT 4 LECTURE 2**CRYSTAL DIRECTIONS**• To find the Miller indices of a direction, • Choose a perpendicular plane to that direction. • Find the Miller indices of that perpendicular plane. • The perpendicular plane and the direction have • the same Miller indices value. • Therefore, the Miller indices of the perpendicular • plane is written within a square bracket to • represent the Miller indices of the direction like [ ]. PH 0101 UNIT 4 LECTURE 2**IMPORTANT DIRECTIONS IN CRYSTAL**PH 0101 UNIT 4 LECTURE 2**DESIRABLE FEATURES OF MILLER INDICES**• The angle ‘’ between any two crystallographic directions • [u1 v1 w1] and [u2 v2 w2] can be calculated easily. The • angle ‘’ is given by, • The direction [h k l] is perpendicular to the plane (h k l) PH 0101 UNIT 4 LECTURE 2**DESIRABLE FEATURES OF MILLER INDICES**• The relation between the interplanar distance and the • interatomic distance is given by, • for cubic crystal. • If (h k l) is the Miller indices of a crystal plane then the • intercepts made by the plane with the crystallographic • axes are given as • where a, b and c are the primitives. PH 0101 UNIT 4 LECTURE 2**SEPARATION BETWEEN LATTICE PLANES**• Consider a cubic crystal of side ‘a’, and a • plane ABC. • This plane belongs to a family of planes • whose Miller indices are (h k l) because • Miller indices represent a set of planes. • Let ON =d, be the perpendicular distance of • the plane A B C from the origin. PH 0101 UNIT 4 LECTURE 2**SEPARATION BETWEEN LATTICE PLANES**PH 0101 UNIT 4 LECTURE 2**SEPARATION BETWEEN LATTICE PLANES**• Let 1, 1 and 1 (different from the interfacial • angles, and ) be the angles between co- • ordinate axes X,Y,Z and ON respectively. • The intercepts of the plane on the three axes are, • (1) PH 0101 UNIT 4 LECTURE 2**SEPARATION BETWEEN LATTICE PLANES**From the figure, 4.14(a), we have, (2) From the property of direction of cosines, (3) Using equation 1 in 2, we get, PH 0101 UNIT 4 LECTURE 2**SEPARATION BETWEEN LATTICE PLANES**Using equation 1 in 2, we get, (4) Substituting equation (4) in (3), we get, PH 0101 UNIT 4 LECTURE 2**i.e.**(5) i.e. the perpendicular distance between the origin and the 1st plane ABC is, PH 0101 UNIT 4 LECTURE 2**Now, let us consider the next parallel plane.**• Let OM=d2be the perpendicular distance of this • plane from the origin. • The intercepts of this plane along the three axes are PH 0101 UNIT 4 LECTURE 2**SEPARATION BETWEEN LATTICE PLANES**• Therefore, the interplanar spacing between two • adjacent parallel planes of Miller indices (h k l ) is • given by, NM = OM – ON • i.e.Interplanar spacing • (6) PH 0101 UNIT 4 LECTURE 2**CRYSTAL SYMMETRY**• Crystals have inherent symmetry. • The definite ordered arrangement of the faces • and edges of a crystal is known as `crystal • symmetry’. • It is a powerful tool for the study of the internal • structure of crystals. • Crystals possess different symmetries or • symmetry elements. PH 0101 UNIT 4 LECTURE 3**CRYSTAL SYMMETRY**What is a symmetry operation ? • A `symmetry operation’ is one, that leaves • the crystal and its environment invariant. • It is an operation performed on an object or pattern • which brings it to a position which is absolutely • indistinguishable from the old position. PH 0101 UNIT 4 LECTURE 3**CRYSTAL SYMMETRY**• The seven crystal systems are characterised by • three symmetry elements. They are • Centre of symmetry • Planes of symmetry • Axes of symmetry. PH 0101 UNIT 4 LECTURE 3**CENTRE OF SYMMETRY**• It is a point such that any line drawn through it will • meet the surface of the crystal at equal distances • on either side. • Since centre lies at equal distances from various • symmetrical positions it is also known as `centre • of inversions’. • It is equivalent to reflection through a point. PH 0101 UNIT 4 LECTURE 3**CENTRE OF SYMMETRY**• A Crystal may possess a number of planes or • axes of symmetry but it can have only one centre • of symmetry. • For an unit cell of cubic lattice, the point at the • body centre represents’ the `centre of • symmetry’ and it is shown in the figure. PH 0101 UNIT 4 LECTURE 3**CENTRE OF SYMMETRY**PH 0101 UNIT 4 LECTURE 3**PLANE OF SYMMETRY**• A crystal is said to have a plane of symmetry, when • it is divided by an imaginary plane into two halves, • such that one is the mirror image of the other. • In the case of a cube, there are three planes of • symmetry parallel to the faces of the cube and six • diagonal planes of symmetry PH 0101 UNIT 4 LECTURE 3**PLANE OF SYMMETRY**PH 0101 UNIT 4 LECTURE 3**AXIS OF SYMMETRY**• This is an axis passing through the crystal such that if • the crystal is rotated around it through some angle, • the crystal remains invariant. • The axis is called `n-fold, axis’ if the angle of rotation • is . • If equivalent configuration occurs after rotation of • 180º, 120º and 90º, the axes of rotation are known as • two-fold, three-fold and four-fold axes of symmetry • respectively. PH 0101 UNIT 4 LECTURE 3**AXIS OF SYMMETRY**• If equivalent configuration occurs after rotation of 180º, • 120º and 90º, the axes of rotation are known as two- • fold, three-fold and four-fold axes of symmetry . • If a cube is rotated through 90º, about an axis normal to • one of its faces at its mid point, it brings the cube into • self coincident position. • Hence during one complete rotation about this axis, i.e., • through 360º, at four positions the cube is coincident • with its original position.Such an axis is called four-fold • axes of symmetry or tetrad axis. PH 0101 UNIT 4 LECTURE 3