Unit V - Crystal Physics PDF

UNIT - V CRYSTAL PHYSICS Introduction 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 us...

UNIT - V CRYSTAL PHYSICS Introduction 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. Classification of Solids Materials differ in their properties due to its crystal structure, based on which solids are classified in to two types. i. Crystalline materials - 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. 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. ii.Amorphous materials - 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., Crystallographic terms i. Crystal - Three dimensional solid which consists of a periodic arrangement of atoms in three dimensional space. ii. Space lattice - A lattice is a regular and periodic arrangement of points in three dimension. It is defined as an infinite array of points in three dimension in which every point has surroundings identical to that of every other point in the array. The Space lattice is also called as Crystal lattice. iii. Basis - A crystal structure is formed by associating every lattice point with an unit assembly of atoms or molecules identical in composition, arrangement and orientation. This unit assembly is called the `basis’. When the basis is repeated with correct periodicity in all directions, it gives the actual crystal structure. The crystal structure is real, while the lattice is imaginary. iv. Unit cell - A unit cell is the smallest unit which, when repeated in space indefinitely, will generate the space lattice. Crystallographic axes Consider a unit cell consisting of three mutually perpendicular edges OA, OB and OC as shown in figure 5.1. Draw parallel lines along the three edges. These lines are taken as crystallographic axes and they are denoted as X, Y and Z axes. Fig. 5. 1 1 Lattice parameters Consider the unit cell as shown in figure 5.2. 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. 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 figure 5.2. These angles ,  and  are called as interaxial angles or interfacial angles. To represent a lattice, the three interfacial angles and their corresponding intercepts Fig. 5. 2 are essential. These six parameters are said to be lattice parameters. Primitive cell - It consists of only one full atom. 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 called as non-primitive cell. Example for primitive cell : Simple Cubic unit cell. Examples for non-primitive cell: BCC and FCC unit cell. SEVEN CRYSTAL SYSTEMS 1. Cubic (isometric) 2. Tetragonal 3. Orthorhombic 4. Monoclinic 5. Triclinic 6. Rhombohedral 7. Hexagonal S. Possible Crystal system Axial lengths Interfacial angles Example No lattice 1 Cubic a=b=c α = β = γ = 90 P, I, F Fe, Cu, Nacl Ordinary 2 Tetragonal a=b≠c α = β = γ = 90 P, I white tin, Indium 3 Orthorhombic a≠b≠c α = β = γ = 90 P, I, F, C Topaz, Sulphur FeSo4, 4 Monoclinic a≠b≠c α = β = 90 γ ≠ 90 P, C Na2So4 CuSo4, 5 Triclinic a≠b≠c α ≠ β ≠ γ ≠ 90 P K2Cr2O7 6 Rhombohedral a=b=c α = β = γ ≠ 90 P Calcite Quartz, 7 Hexagonal a=b≠c α = β = 90 γ = 120 P Tourmaline 2 CRYSTAL SYSTEMS Crystals are classified into seven crystal systems on the basis of lattice parameters such as axial lengths a, b, c and interfacial angles α, β, γ. They are 1. Cubic (isometric) 2. Tetragonal 3. Orthorhombic 4. Monoclinic 5. Triclinic 6. Rhombohedral 7. Hexagonal. 1.CUBIC SYSTEM Here, three crystal axes are perpendicular to each other and axial lengths are same along all the three axis. Lattice paramters are a = b = c, α = β = γ = 90◦ Ex: Iron, Copper 2. TETRAGONAL SYSTEM Here , three crystal axes are perpendicular to each other. Two of the three axial lenghts are the same, but the third axial length is different. Lattice paramters are a = b ≠ c, α = β = γ = 90◦ Ex: White tin, Indium 3. ORTHORHOMBIC SYSTEM Here, three crystal axes are perpendicular to each other. All the three axial lengths are of unequal lengths. Lattice paramters are a ≠ b ≠ c, α = β = γ = 90◦ Fig. 5.3 (a) Ex: Sulphur, Topaz 4. MONOCLINIC SYSTEM Here, two of the crystal axes are perpendicular to each other, but the third is obliquely inclined.Three axial lengths are different. Lattice paramters are a ≠ b ≠ c, α = β = 90◦γ ≠ 90◦ Ex: Sodium sulphite, Fe3SO4. 5. TRICLINIC SYSTEM Here, two of the crystal axes are not perpendicular to each other, axial lengths are also not equal. Lattice paramters are a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90◦ Ex: Copper sulphate. 6. RHOMBOHEDRAL SYSTEM Here, the three crystal axes are inclined to each other at an angle other than 90◦, three axial lengths are equal along the axes. Lattice paramters are a = b = c, α = β = γ ≠ 90◦ Ex: Calcite 7. HEXAGONAL SYSTEM Here, two of the crystal axes are 120◦ apart, while the third is perpendicular to both of them. The axial lengths are the same along the axes that are 120◦ apart, but axial length along the third axis is different. Lattice paramters are a = b ≠ c, α = β = 90◦, γ = 120◦ Ex: Quartz, Tourmaline 3  BRAVAIS LATTICES According to Bravais, there are 14 possible ways of arranging points in space such that environment looks same from each point. 1.CUBIC LATTICE  Simple cubic has 8 corner atoms, 1 at each corners.  BCC has 8 corner atoms 1 at each corners and another atom at the centre of the body.  FCC has 8 corner atoms 1 at each of 8 corners. In addition there are 6 face centred atom 1 at each of the 6 faces of the cubic. 2.TETRAGONAL LATTICE  Simple tetragonal has 8 corner atoms, one at each corners.  Body centred tetragonal has 8 corner atoms and 1 at the body centre. 3. ORTHORHOMBIC LATTICE  Simple orthorhombic has 8 corner atoms.  Body centred orthorhombic there are 8 corner atoms and one atom at body centre.  Face centred orthorhombic has 8 corner atoms and 6 face centred atoms.  Base centred orthorhombic has 8 corner atoms and 2 atoms at the centre of 2 faces. 4. MONOCLINIC LATTICE  Simple monoclinic has 8 corner atoms, one at each corners.  Base centred monoclinic has 8 corner atoms, one at each corners and 2 atoms at the centre of faces. 5.TRICLINIC LATTICE  Simple monoclinic has 8 corner atoms, one at each corners. 6.RHOMBOHEDRAL LATTICE  Simple rhombohedral has 8 corner atoms, one at each corners. 7.HEXAGONAL LATTICE Fig. 5.3 (b)  Simple hexagonal has 12 corner atoms, one at each corners and 2 atoms at the centre of 2 hexagonal faces. Characteristics of an unit cell i) No. of atoms per unit cell (n) - Number of atoms possessed by a unit cell ii) Coordination number (CN) - Number of nearest atoms directly surrounding a particular atom in a crystal. iii) Nearest neighbouring distance (a) - Distance between the centers of two nearest neighbouring atoms (2r). iv) Atomic radius (r) - Half of the nearest neighbouring distance in a crystal. v) Atomic packing factor (APF) - Ratio of the volume of atoms in the unit cell to the volume of the unit cell. There is no unit for it since it is a ratio. 4 SIMPLE CUBIC (SC) STRUCTURE In this structure, there is one atom at each of the 8 corners of unit cell. These atoms touch each other along cubic edges. No. of atoms per unit cells It is the number of atoms possessed by the unit cell. There is one atom at each corner of unit cell. Each corner atoms is shared by 8 adjacent unit cells. 1 1 Fig. 5.4 (a) Share for one unit cell  No. of atoms / unit cell   8  1 atom 8 8 Total number of atoms in a unit cell = 1 atom Coordination number It is the number of nearest atoms directly surrounding a particular atom in a crystal.Simple cubic unit cell has 8 corner atoms. Let us consider one of the corner atom (X). It is in direct touch with 6 adjacent unit cells edge atoms. Hence, coordination number for simple cubic is 6. Fig. 5.4 (b) Atomic Radius It is half of the distance between two nearest neighboring atoms in a crystal. Consider a face of unit cell of SC. The atoms touch each other along the edges. If ‘a’ is the side of the unit cell and ‘r’ its radius, then 2r  a ; r  a 2 Packing factor Fig. 5.4 (c) It is defined as the ratio of total volume occupied by the atoms in the unit cell (v) to the volume of unit cell (V). 𝑁𝑜.𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑡 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 ×𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑜𝑛𝑒 𝑎𝑡𝑜𝑚 P.F = 𝑡𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑛 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 Number of atoms per unit cell = 1 4 Volume of one atom   r3 3 Total volume of unit cell = a3 4 1  r 3 P.F. =  3 a3 a By substituting r  2 3 a 4   2 4  a 3  P.F.    = 0.52×100 3 a 3 3 8 a 3 6 P.F. = 52 % Thus 52 % volume of unit cell is occupied by the atoms and remaining 48 % volume is vacant. Ex:- Polonium. 5 BODY CENTERED CUBIC (BCC) STRUCTURE In this structure, the unit cell has one atom at each corner and one atom at the body centre of the cube. Number of atoms per unit cell It is the number of atoms possessed by the unit cell. Each corner atom is shared by 8 surrounding unit cells. 1 Fig. 5. 5 (a) No. of corner atoms / unit cell   share for one unit cell 8 1 Contribution due to corner atoms   8  1 atom 8 There is one atom at the body centre of each unit cell Total number of atoms per unit cell = 1+1= 2 atoms Fig. 5. 5 (b) Coordination number It is the number of nearest atoms directly surrounding a particular atom in a crystal. In BCC structure, the corner atoms do not touch each other. But all ‘8’ corner atoms touch the body centre atom along the body diagonal. If body centered atom is a reference atom, there are ‘8’ nearest neighbours. Hence, the coordination number of BCC structure is ‘8’. Atomic radius It is half of the distance between two nearest neighboring atoms in a crystal.Consider the atoms at A, G and O. These atoms lie in a straight line along the body diagonal. From the figure, Fig. 5. 5 (c) AG = r+2r+r = 4r  AG2 = (4r)2 = 16 r2 (1) From the  ABC, AC2 = AB2+BC2 = a2+a2 = 2a2 From the  ACG, AG2 = AC2+CG2 = 2a2+a2 AG2= 3a2 (2) Using (2) in (1), 3a2 = 16r2 3a 2 3a r2  ; r 16 4 Packing factor It is defined as the ratio of total volume occupied by the atoms in the unit cell (v) to the volume of unit cell (V) v P.F  V 6 Number of atoms per unit cell = 2 4 Volume of 1 atom in the unit cell   r3 3 3a Atomic radius  4 Volume of the unit cell = a3 4 2  r 3 P.F  3 a3 By substituting r, 3  3a  2  4     4  P.F  3 a 3 8  3 3a 3  3 a3  4  4  4 3  = 0.68×100 ; P.F = 68 % 8 Thus packing factor is 68 %, which means 68 % of the volume of unit cell is occupied by atoms and the remaining 32 % volume is vacant. Ex:- Tungsten, Chromium FACE CENTERED CUBIC (FCC) STRUCTURE In FCC structure, the unit cell has one atom at each corner of the cube and one atom at the centre of each face. Number of atoms per unit cell It is the number of atoms possessed by the unit cell. There are 8 corner atoms, one at each of its 8 corners. Each corner atom is shared by 8 surrounding unit cells. 1 Share of each unit cell  of corner atoms 8 Fig. 5. 6 (a) 1 Number of atoms per unit cell due to corner atom   8 =1 atom 8 Each face centre atom is shared by 2 adjacent unit cells. 1 Hence number of atoms per unit cell due to face center atom   6 =3 atom 2 Total number of atoms per unit cell = 1+3 =4 atoms Coordination number It is the number of nearest atoms directly surrounding a particular Fig. 5. 6 (b) atom. Consider three mutually perpendicular planes with a common corner atom. In each plane of I, II & III, there are four face centered atoms touch with common corner atom. CN = 4+4+4 = 12 Therefore, coordination number is 12. 7 Atomic radius It is half of the distance between two nearest neighbouring atoms. Consider a face of ABCD in FCC structure. In which Face centered atom is touch with four corner atoms along the face diagonal. AC2 = AB2+BC2 AC2 = a2+a2 AC2 = 2a2 (1) Interms of atomic radius, AC = 4r AC2 = (4r)2 = 16 r2 (2) 2 2 Using (1) & (2), 2a = 16r 2a 2 a 2 r2   16 8 a a r  (or) r =(√2 a)/4 8 2 2 Packing Factor Fig. 5. 6 (c) It is defined as the ratio of total volume occupied by the atoms in a unit cell to the total volume of a unit cell. v P.F  V Number of atoms per unit cell = 4 4 Volume of 1 atom   r3 3 a Atomic radius  2 2 Volume of unit cell = a3 3  a    4  r 3 16   2 2  4 P.F  3    3 3 a 3a 16  a 3  3 3a  8  2 2   = 0.74×100 3 2 P.F. = 74 % Thus the packing factor is 74 % (ie) 74 % of the volume of unit cell is occupied by the atoms and the remaining 26 % of volume is vacant. Ex:- Copper, Aluminum. HEXAGONAL CLOSE-PACKED (HCP) STRUCTURE  The unit cell has one atom at each of 12 corners of the hexagonal face, one atom each at the centre of the two hexagonal faces.  3 atoms symmetrically arranged inside the unit cell. 8   There are 3 layers of atom in it.  At the bottom layer, the central atom has 6 nearest neighbouring atoms in the same plane.  The middle layer at a distance of C/2 from the bottom layer has 3 atoms.  The bottom layer is similar to top layer which is at a distance of ‘C’. Number of atoms per unit cell: It is the number of atoms possessed by the unit cell. Each corner is shared by 6 surrounding unit cells. Hence number of atoms per unit cell due to corner atom 1   12  2 atoms Fig. 5. 7 (a) 6 Each face centered atom is surrounded by 2 unit cell 1 Number of atoms per unit cell due to face centered atom   2  1 atom 2 There are 3 atoms completely inside the unit cell which are not shared by any other adjacent unit cell. Total number of atoms per unit cell = 2+1+3 = 6 atoms 2. Coordination number:  Consider the bottom layer the central atom has 6 nearest neighbouring atoms in the same plane. Fig. 5. 7 (b)  At a distance C/2 from the bottom layer, there are two layers, one above and other below the bottom layer containing 3 atoms in each layer are also neighbouring atoms. In total there are 3+6+3 = 12 nearest neighbouring atoms. Coordination number is 12. 3. Atomic radius: It is half of the distance between two nearest neighbouring atoms. Here atoms are touch each other along the edges of the hexagon. The nearest neighbouring distance a = 2r. a Atomic radius, r  2 4. Packing Factor: It is defined as the ratio of total volume occupied by the atoms in a unit cell to the total volume of a unit cell. 𝑁𝑜.𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑝𝑒𝑡 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 ×𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑜𝑛𝑒 𝑎𝑡𝑜𝑚 P.F = 𝑡𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑛 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 Fig. 5. 7 (c) Number of atoms per unit cell = 6 4 Volume of one atom   r3 3 a Atomic radius, r  2 9 Calculation of c/a ratio: Consider a triangle ABO in the bottom layer. Above these ABO atoms at a perpendicular distance C/2, the next layer atom lies at point ‘c’. AY  ABO, a line AY is drawn perpendicular to BO, cos 30 o  AB a 3 AY = AB cos 30 o  2 X is a centre of  ABO, then 2 a 3 AX =  3 2 a AX = 3 In the  ABO AC2 = AX2+CX2 a c Substituting the values for AC = a, AX = , CX = 3 2 2  a  c 2 a2 =       3  2 a2 c2 a2 =  3 4 c2 a  a2  4 3 c 2 2a 2  4 3 Fig. 5. 7 (d) 2 c 8  a 3 c 8  a 3 Calculation of volume of unit cell: The volume of unit cell can be calculated by multiplying the base area with the height of the unit cell. The base area can be calculated by multiplying 6 with area of one equilateral triangle. 3a 2 Area of one equilateral triangle  4 3a 2 Volume of unit cell  6  c 4 3 3a 2 c V  2 3 4 a 6  r 3 24   2 P.F. = 3  2 2 3 3a c 3  3 3a 2 c 2 10  24  a 3  c = 9 3a 2 c  8 2 a P.F = 3 3c By substituting c/a value 2 3 2 3 P.F = = 3 3 8 3 32 2  = = 0.74×100 3 2 Packing factor = 74 % Therefore, 74 % of volume of unit cell is occupied by the atoms and the remaining 26 % volume is vacant. DIAMOND CUBIC (DC) STRUCTURE This structure is a combination of two interpenetrating FCC sub lattices. Number of atoms per unit cell It is the number of atoms possessed by the unit cell. In the unit cell of diamond, the carbon atoms are present at three different positions of the unit cell. (i) Corner atoms (ii) Face centered atoms (iii) Four atoms present fully inside unit cell. Fig. 5. 8 (a) Each corner atom is shared by 8 unit cells. 1 Number of atoms due to corner atoms per unit cell   8 =1 atom 8 1 Number of atoms per unit cell due to face centered atom   6 =3 atom 2 Inside unit cell, 4 atoms which are not shared by any other surrounding unit cells. Total number of atoms per unit cell = 1+3+4 = 8 atoms Atomic radius The corner atoms do not touch each other. But both face centered atoms and corner atoms touch with the 4 atoms situated inside the unit cell.The atoms X and Y are touch each other along the body diagonal with the position X(0,0,0) and Y (a/4,a/4,a/4). Fig. 5. 8 (b) A perpendicular is drawn from ‘Y’ atom which meets the unit cell at a point ‘Z’ which is at a distance of a/4. 11  XY2 = XZ2+ZY2 XY2 = XT2+TZ2+ZY2 2 2 2 a a a       4 4 4 3a 2 3a XY2  XY  16 ; 4 W.K.T, XY = 2r Fig. 5. 8 (c) 3a  r 8 Coordination number It is half of the distance between two nearest neighbouring atoms. In this structure, inner atom (1) is touch with 3 nearest face centered atom and one corner atom. So the coordination number for diamond = 4 Packing Factor It is defined as the ratio of total volume occupied by the atoms in a unit cell to the total volume of unit cell. No.of atoms per unit cell ×volume of one atom P.F = 𝑡𝑜𝑡𝑎𝑙 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎𝑛 𝑢𝑛𝑖𝑡 𝑐𝑒𝑙𝑙 Number of atom per unit cell = 8 4 Volume of one atom   r3 3 3 Volume of unit cell = a 4 8  r 3 P.F  3 a3 3  3a  8  4     8  8  4  3  3  a 3   3 a3 3 a3  8  8  8  3  = 0.34×100 16 Thus 34 % volume of unit cell is occupied by atoms and the remaining 66 % volume is vacant. MILLER INDICES Definition Miller indices are defined as the reciprocal of the intercepts made by the plane on the crystallographic axes which are reduced to smallest numbers. Procedure for finding Miller Indices: Let us consider a plane ABC which cuts 1unit along x–axis, 2 unit along y–axis, and 3 unit along z-axis. Step 1 Find the intercepts made by the plane ABC along the three axis interms of axial length. 12  OA:OB:OC = pa:qb:rc [p =1,q = 2, r = 3]  The intercepts are 1a : 2b: 3c Step 2 Find the coefficients of intercepts, p, q, r (i.e) 1,2,3 Step 3 1 1 1 Find the reciprocal of these numbers, , , p q r 1 1 1 (i.e) , , 1 2 3 Step 4 Convert these reciprocals into whole numbers by multiplying each Fig. 5. 9 and every reciprocal with their least common multiplier. 1 1 1 Since 6 is the LCM in this case, we get 6  ,6  ,6  1 2 3 We have 6 3 2 Step 5 Enclose these numbers in a bracket.i.e. (6 3 2). This is called Miller Indices of the plane. The general notation for Miller Indices is (h k l). Significance of Miller Indices  The Miller Indices should be enclosed only in this bracket ( ).  There should not be any commas in between the numbers.  If the Miller Indices are say (6 3 2) means it should be read as six three two.  The direction of plane can be represented by enclosing the Miller Indices in a square bracket [h k l]. IMPORTANT FEATURES OF MILLER INDICES  If a plane is parallel to any one of the coordinate axes , the corresponding intercept will be infinite (∞) and therefore the corresponding Miller Index is zero  Miller indices don’t define a particular plane but a set of equidistant parallel planes. They are represented as < h k l > a b c  For a group of parallel planes < h k l > the intercepts can be represented as , , h k l  A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts.  If a plane has negative intercept the corresponding Miller index will be written with a bar. Ex. (1 1 1 ) Relation between Interplanar spacing, Lattice constant & Miller Indices (d-Spacing) It is perpendicular distance between successive parallel planes. Consider a plane ABC in Cubic crystal with ‘a’ as length of cube edge and Miller Indices of (h k l) 13  A normal ON is drawn to the plane ABC from the origin plane. Let ON be the interplanar spacing (d). This plane ABC makes the intercepts OA, OB, OC on the crystallographic axes X, Y, Z.  ' ,  ' and ' are the angles between the axes X, Y, Z and the normal ON respectively. 1 1 1 OA: OB: OC = : : h k l By multiplying by lattice constant ‘a’ a a a OA: OB: OC = : : h k l a a a OA = , OB = , OC = h k l From the  OAN ON d hd cos '    OA a a h Fig. 5. 10 (a) From the  OBN ON d kd cos  '    OB a a k From the  OCN ON d ld cos  '    OC a a l The law of direction cosines is cos 2  ' cos 2  ' cos 2  '  1 2 2 2  hd   kd   ld        1  a   a  a  d2 2 2 h  k2  l2 1  a a2 d2   h2  k 2  l 2  Fig. 5. 10 (b) a d  h 2  k2  l2  MILLER INDICES FOR DIFFERENT PLANES Fig. 5. 11 14 GRAPHITE STRUCTURE  Graphite, one of the allotropic forms of carbon.  Carbon atoms arranged in regular hexagons in flat parallel layers such that each carbon atom is linked by the neighboring atoms.  When each carbon atom forms three covalent bonds with three other carbon atoms in the same plane with bond length 1.42 Ȧ, sheets of graphite are produced.  The sheets are held together in a crystal by Vander walls force with spacing about 3.4 Ȧ.  It shows that there is no strong binding between different layers.  Therefore layers are easily separable from each other.  The weak intersheet bonding explains the softness of graphite and hence it used as a lubricant.  The fourth bonding electron of carbon is delocalized and resonates between three covalent bonds due to this graphite have good electrical and thermal conductivities. Fig. 5. 12 PROPERTIES  High thermal conductivity  High thermal shock resistance  High melting temperature  Low density  Low hardness  High electrical conductivity APPLICATIONS  As a lubricant at higher temperature  For manufacturing lead pencils and stove paints  It is a component of printer’ ink 15  CRYSTAL IMPERFECTIONS If atoms in the solid are not arranged in a perfectly regular manner, it is called defects (or) imperfections in crystals. It is observed that the crystals are rarely found to be perfect. The effect of imperfections is found to be very important in understanding the properties of crystals. This is because, structure insensitive properties like stiffness, density and electrical conductivity are not affected much by the presence of imperfections of defects in the crystals. Structure sensitive properties such as, mechanical strength, ductility, crystal growth, magnetic hysteresis, dielectric strength which are of greatest technical importance are found to be greatly affected by the presence of imperfections. Various types of Crystal Imperfections 1. Lattice vibrations or Phonons 2. Point defects or zero dimensional defects a) Vacancies: Frenkel & Schottky b) Interstitial defects: Self & Foreign c) Impurities: Substitutional & Interstitial 3. Line defects or one dimensional defects or dislocations a) Edge dislocation b) Screw dislocation c) Dislocation climb d) Dislocation slip. 4. Surface defects or plane defects or two dimensional defects a) Grain boundaries b) Twin & Twist boundaries c) Tilt boundaries d) Stacking faults 5. Volume defects of three dimensional defects or bulk defects. Lattice vibrations or Thermal Vibrations or Phonons Mathematically, atoms in a perfect lattice should occupy exactly the sites to which they are attached. No crystal is perfectly rigid, since it can be deformed by finite forces. It is possible to displace the atoms from their ideal sites with a finite expenditure of energy due to lattice vibrations (thermal vibrations). The frequency of vibration is almost independent of temperature, but the amplitude increases with increasing temperature. When the vibrations become strong enough the atoms may break the bonds between them. Since the atoms interact with one another they tend to vibrate in synchronism. The thermal vibration of atoms of a solid does not seriously disturb the perfect structure of the crystal. 16 CRYSTAL IMPERFECTIONS The disturbance occurred in the regular orientation of atoms is called crystal defect or imperfection. Classification of crystal defects  Point defect – Zero dimensional  Line defect – One dimensional  Surface defect – Two dimensional  Volume defect – Three dimensional Fig. 5. 13 (a) 1) Point defect: Point defects are crystalline irregularities in one or two atomic dimension. There are three types of point defects. a) Vacancies One or more atoms are missing from a normal occupied position is known as vacancy. There are different kinds of vacancies like Frenkel defect, Schottky defect etc. i) Schottky defect Two oppositely charged ions are missing from an ionic crystal. This is known as Schottky defect. Since a pair is missing, electrical neutrality is maintained. ii) Frenkel Defect A vacancy associated with interstitial impurity is called Frenkel defect. Frenkel defect does not change the overall electrical neutrality. b) Interstitial defect When an extra atom occupied interstitial space within the crystal structure without removing parent atom, the defect is called interstitial defect. There are two types namely self and foreign interstitial. i) Self Interstitial If an atom from same crystal occupies interstitial site, then it is called self Interstitial. Fig. 5. 13 (b) ii) Foreign Interstitial If an impure atom occupies interstitial site, then it called foreign interstitial. c) Impurities When the foreign atoms are added to crystal lattice, they are known as impurities. The defect is called impurity defect. i) Substitution impurity A substitutional impurity refers to foreign atoms that replace a parent atom in the lattice. ii) Interstitial impurity An interstitial impurity is a small size atom occupying the empty space in the parent crystal, without dislodging any of the parent atoms from their sites. 2) Line defects The defects due to dislocation or distortion of atoms along a line are known as line defects. These defects are also called dislocation. Burger's Vector The vector which indicates the direction and magnitude of the shift of the lattice on the slip plane is called Burger's Vector.. 17 Types of dislocations i) Edge dislocation An edge dislocation arises when one of the atomic planes forms only partially and does not extend throughout the lattice. The atomic plane AB terminates at B. It is viewed as an extra plane inserted in between a set of parallel planes. This is called edge dislocation. There are two types of edge dislocations namely Positive edge dislocation () & Negative edge dislocation (T). If the extra plane of atoms above the slip Fig. 5. 13 (c) plane than that edge dislocation is called positive. If the extra plane of atom below the slip plane than that edge dislocation is called negative. i) Screw dislocation Screw dislocation is due to a displacement of atoms in one part of a crystal relative to rest of the crystal. This dislocation forms a spiral ramp around dislocation line.If spiral motion of one part of the crystal is in clockwise direction then dislocation is right handed, on the other hand, spiral motion is in anti-clockwise directions then the dislocation is left handed. Fig. 5. 13 (d) 3) Surface defects The defects on the surface of material are called surface defects.The surface defects are Grain boundaries, Tilt & twist boundaries, and Twin boundaries, Stacking fault i) Grain boundaries Whenever the grains of different orientations separate the general pattern of atoms and exhibit a boundary, the defect caused is called grain boundary. A grain boundary is formed when two growing grain surfaces meet. The shape of the grain is usually influenced by the presence of Fig. 5. 13 (e) surrounding grains. 18 i) Tilt and twist boundaries Tilt boundaries are an array of parallel edge dislocation of same sign arranged one above other in an array or series. Twist boundaries consists of atleast two set of parallel screw dislocation lying in the boundary. ii) Twin boundaries If the boundaries in which the atomic arrangement on one Fig. 5. 13 (f) side of the boundary is somewhat a mirror image of the other side, then that defect is called twin boundary. 3. Stacking faults Whenever the stacking of atoms is not in proper sequence throughout the crystal, defect caused is called stacking fault. Fig. 5. 13 (g) Fig. 5. 13 (h) ROLE OF IMPERFECTIONS IN PLASTIC DEFORMATION Plastic deformation is defined has the process of permanent deformation which exists in a metal even after the removal of the stress. This occurs due to rolling, forging, drawing, spinning etc., The plastic deformation in crystalline materials occurs at low temperature where the application of stress is very small. Plastic deformation usually occur under the tensile, compressive or torsional stresses. There are two basic modes of plastic deformation namely slip or gliding and twinning. The slip mode is commonly found in many crystals at elevated temperature where the twinning at occurs at lower temperature. Modes of Plastic deformation: (1). Slip or gliding The shear deformation in which the atom moves in one crystal plane over the atom of another crystal plane by inter atomic distance relative to its initial position. Lower stresses is enough to produce slip. It occurs due to the movement of dislocation. The orientation of the crystal above and below the slip plane is same before and after deformation. Slip band is formed after several milli seconds. (2). Twinning The shear deformation in which a portion of the crystal takes up another portion as a mirror image. Higher stresses is required. It occurs due to mechanical deformation. Twinning results in an orientation difference across twin plane. Twins can be formed in few micro seconds. 19 CRYSTAL GROWTH - SOLUTION GROWTH It is a simplest method to grow many crystalline materials which are highly soluble. Low temperature solution growth: The low temperature solution growth is suitable for materials which decompose at high temperature and undergo phase transformation below the melting point. There are two methods of low temperature solution growth. They are  Slow cooling method  Slow evaporation method SLOW COOLING METHOD Slow cooling is the easiest method to grow bulk single crystals from solution. This technique needs only a vessel for the solution, in which the crystals grow. The temperature at which crystallization begin is in the range of 45 C to 75 C and the lower limit of the cooling is the room temperature. SLOW EVAPORATION METHOD In this method, the saturated solution is kept at a particular temperature and provision is made for evaporation. This is simple and convenient method of growing single crystals of large size.  Growth of strain and dislocation free crystals.  Permits the growth of prismatic crystals by varying the growth conditions  Only method which can be used for substances that undergo decomposition before melting.  A variable rate of evaporation may affect the quality of the crystal. Fig. 5. 14 (a) 20  CRYSTAL GROWTH - CZOCHRALSKI METHOD Principle It is a crystal pulling technique of growth of crystal by a gradual layer by layer condensation of melt. Construction & Working Fig. 5. 14 (b)  The pure material which is to be grown in the form of a single crystal is taken in the crucible.  The material is heated above the melting point by using induction heater. Thus the melt is obtained in the crucible.  A small defect free single crystal, called seed crystal is introduced into the melt by means of a crystal holder.  A small portion of seed crystal is initially melted. The temperature is adjusted for the growth of seed crystal by the solidification of melt into a thin layer of crystal.  The seed crystal is rotated and gradually pulled out of the melt by maintaining the grown crystal and melt interface near the surface of the melt.  Thus a single crystal grown is on the seed crystal. The diameter of the grown crystal is controlled by the temperature of the melt and rate of pulling. Advantages 1. It will produce strain free surface. 2. It can produce large single crystal called ingot. Limitations 1. High vapour pressure of the materials can be produced. 2. It may produce contamination of melt by the crucible. 21 CRYSTAL GROWTH - BRIDGMANN TECHNIQUE Principle The selective cooling of the molten material to form single crystal by solidification along a particular direction is the principle of Bridgmann Technique. Construction & Working  In this method the material to be grown in the form of single crystal is taken in a cylindrical sealed crucible.  The crucible is made of platinum and tapered conically with pointed tip at the bottom.  The crucible is suspended in the upper furnace until the material in the crucible is completely melted. Fig. 5. 14 (c)  The crucible is then slowly lowered from upper furnace into lower furnace with help of an electric motor.  The temperature of the lower furnace is maintained below the melting temperature of the melt.  Since the pointed tip enters the lower furnace first, the melt at this point starts to solidify to form crystal.  As the crucible is continuously lowered, the solidification of melt continues to form crystal until all the melt become a solid crystal. Advantages 1. It is a simple technique. 2. It enables easy stabilization of temperature gradients. 3. Crucible can be evacuated and sealed. Hence oxidation of melt is prevented. Limitations 1. Confinement of crystal may occur. 2. No visibility of material during growth. 22 EPITAXIAL GROWTH: Epitaxy refers to the method of depositing a mono-crystalline film on a mono- crystalline substrate. The deposited film denoted as epitaxial film or epitaxial layer. Epitaxial films may be grown from gaseous or liquid precursors. The substrate acts as a seed crystal, the deposited film takes on a lattice structure and orientation identical to those of the substrate. Hence the quality of single crystalline growth depends on the crystallinity of substrate. TYPES OF EPITAXIAL FILMS: Epitaxial films can be classified into two broad categories: Based on material Homoepitaxy: The film and substrate are the same material. Heteroepitaxy: Film and substrate are of different materials. Based on growth methods Liquid Phase Epitaxy: The film grown using liquid solutions over a substrate termed as liquid phase epitaxy (LPE). Vapour Phase epitaxy: The film grown by depositing vapour of materials over a substrate is termed as vapour phase epitaxy. Molecular beam epitaxy: A thin single crystal layer deposited on a single crystalline substrate using atomic or molecular beams generated in Knudsen cells contained in an ultra-high vacuum chamber (10-8–10-12 Torr) is termed as Molecular beam epitaxy (MBE). LIQUID PHASE EPITAXY (LPE) Liquid phase epitaxy is a method to grow semiconductor crystal layers from the melt on solid substrates. Basically, it is a high temperature solution growth technique in which a thin layer of the required material is deposited onto a suitable substrate. Liquid Phase Epitaxy method characterized as a near equilibrium growth process, when compared to the various vapour phase epitaxy techniques. The semiconductor dissolved in the melt of another material. At conditions that are close to the equilibrium between dissolution and deposition of the semiconductor, 23 crystallization takes place on the substrate slowly and uniformly. The equilibrium conditions depend very much on the temperature and on the concentration of the dissolved semiconductor in the melt. A crystalline layer of material is grown from a saturated liquid solution. The process takes place in a high-purity carbon boat, which contains the substrate in the lower part and a series of bins with melted materials in each bin. For example, one bin might hold gallium (Ga), another aluminium (Al), and another gallium arsenide (GaAs). The substrate is placed in contact with each bin in turn, the temperature is reduced, and an exact thickness of material is crystallized. Thus mono-crystalline solid can be produced by liquid phase epitaxy. Generally the growth takes place in the presence of hydrogen atmosphere to achieve oxide free crystals. Fig. 5. 15 ADVANTAGES:  Simple and inexpensive apparatus  More safety and accurate temperature control  Rate of growth of single crystal is more and grown crystal quality is very good.  Near equilibrium growth for precise alloy compositions.  High degree of reproducibility and proper thickness and composition controls. DISADVANTAGES:  Unable to grow immiscible alloys.  The maximum thickness of single crystal layer is less than 1000 A˚  Difficult to grow abrupt heterostructures.  Cannot prepare large no. of epitaxial layers on wafers. 24