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ATOMIC PACKING FACTOR (APF),Crystal structure of sodium chloride (NaCl) Or Rock Salt,MILLER INDICES,X – Rays Diffraction,Laue Experimental demonstration,Bragg reflection,Braggs spectrometer,

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- 1. 1/23/2017 1 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad Engineering Physics II Dr. A K Mishra Associate Professor Applied Science Department Jahangirabad Institute of Technology, Barabanki
- 2. • CRYSTAL STRUCTURE AND X – RAYS DIFFRACTION • DIELECTRIC AND MAGNETIC PROPERTIES OF MATERIALS • DIELECTRIC PROPERTIES • MAGNETIC PROPERTIES • ELECTROMAGNETIC THEORY • PHYSICS OF SOME TECHNOLOGICAL IMPORTANT MATERIALS • SEMICONDUCTOR • SUPERCONDUCTOR • NANO-MATERIALS 1/23/2017 2 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad
- 3. Crystal Structure: Introduction Matter is classified into three kinds Solids : – Atoms or molecules are arranged in a fixed manner – Solids have definite shape and size – On basis of arrangement of atoms or molecules, solids are classified into two categories, they are crystalline solids and amorphous solids. Liquids : – Atoms or molecules are not arranged in a fixed manner – Solids have not definite shape and size Gases: – Atoms or molecules are not arranged in a fixed manner – Solids have not definite shape and size 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 3
- 4. CRYSTALLINE SOLIDS AMORPHOUS SOLIDS In crystalline solids, the atoms or molecules are arranged in a regular and orderly manner in 3-D pattern, called lattice. In amorphous solids, the atoms or molecules are arranged in an irregular manner, otherwise there is no lattice structure. These solids passed internal spatial symmetry of atomic or molecular orientation. These solids do not posses any internal spatial symmetry. If a crystal breaks, the broken pieces also have regular shape. Eg: M.C : Au, Ag,Al, N.M.C: Si, Nacl, Dia. If an amorphous solid breaks, the broken pieces are irregular in shape. Eg : Glass, Plastic, Rubber. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 4 Crystal Structure
- 5. 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. Three Dimensional- 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 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 5 lyrespectiveaxisz,y,x,alongvectorsonaltranslatiare,,and integeresarbitraryare,, 321321 cba nnnwherecnbnanT
- 6. BASIS : • The unit assembly of atoms (molecules or ions) identical in composition ,arrangement and orientation is called basis. In elemental crystals like, Aluminum , barium, copper , silver , sodium etc. The basis is a single atom. • In NaCl, KCl,AgI etc. basis has two atom or basis is diatomic . • In case of CaF2,Sno2,Sio2 etc. the basis has three atoms or basis is Triatomic. • The basis more than two atom is called multi-atomic • Space lattice + Basis = CRYSTAL STRUCTURE. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 6
- 7. • The arrangement within it, when repeated in three dimension gives the total structure of the crystal. • An arbitrary arrangement of crystallographic axis marked X,Y,&Z. • The angles between theThree crystallographic axes • are known as interfacial angles or interaxial angles. • The angle between the axes X and Y= α • The angle between the axes Z and X = β • The angle between the axes Z and Y = γ The intercepts a,b&c define the dimensions of an unit cell and are known as its primitive. The three quantities a,b&c are also called the fundamental translational vectors. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 7 b c α β ϒ a UNIT CELL AND LATTICE PARAMETERS
- 8. Primitive Unit Cell • A primitive cell or primitive unit cell is a volume of space that when translated through all the vectors in a Bravais lattice just fills all of space without either overlapping itself or leaving voids. • A primitive cell must contain precisely one lattice point. • Unit cell drawn with lattice point at each corner but lattice point at the center of certain faces Unit cell with lattice point at corners only ,called primitive cells. • Unit cell may be primitive cell but all primitive cell is not necessarily unit cell. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 8
- 9. SEVEN CRYSTAL SYSTEM AND FOURTEEN BRAVAIS LATTICES 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 9 Crystal system Type of lattice CUBIC SIMPLE,FACE CENTERED,BODY CENTERED TETRAGONAL SIMPLE,BODY CENTERED ORTHOROMBIC SIMPLE,FACE CENTERED,BODY CENTERED END CENTERED MONOCLINIC SIMPLE,END CENTERED RHOMBOHEDRAL SIMPLE TRICLINIC SIMPLE HEXAGONAL SIMPLE
- 10. SYSTEM LATTICE PARAMETER INTERFACIAL ANGLE EXAMPLE CUBIC a=b=c α=β=ϒ=900 Nacl,Kcl,Diamond TETRAGONAL a = b ǂ c α=β=ϒ=900 SiO2 , TiO2 ORTHOROMBIC a ǂ b ǂ c α=β=ϒ=900 KNO3 MONOCLINIC a ǂ b ǂ c α=β=900, ϒǂ900 Na2SO4 , 10H2O RHOMBOHEDRAL a=b=c α=β=900, ϒǂ900 Calcite, NaNO3 TRICLINIC a ǂ b ǂ c α ǂ β ǂ ϒ ǂ 900 CuSO4 , 5H2O HEXAGONAL a = b ǂ c α=β=900, ϒ=1200 Graphite, ZnO 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 10
- 11. Space lattice of cubic system SIMPLE CUBIC CELL (SCC): • More than half elements crystallized into the cubic system (a=b=c, α=β=ϒ=900 ). • In SCC atoms are situated at the corners of the cell such that they are touches each other. • Only one-eight part of atom remains in each cell rest contributing others. • Therefore there is only one atoms per unit cell. • The cell containing only one atom is called primitive cell, SCC is also known as cubic P cell. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 11 SCC
- 12. BODY CENTERED CUBIC CELL (BCC) • Atoms are situated at each corner of the unit cell and also at the intersection of the body diagonal also known as cubic 1 cell. • Each cell has eight corners and eight cell meet at each corner. • One-eight atom contribute to any one cell and one atom at center of each cell. • BCC has two atom per unit cell. • Total No. of atom in any one cell is( x8 +1)=2 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 12 8 1 BCC
- 13. FACE CENTERED CUBIC CELL (FCC) • Atoms are situated at all eight corners of the cell and also at the center of all the six faces of the unit cell. • It is also known as cubic F cell. • One-eight of the atom belongs to any one cell, similarly an atom at the corner of the face shared by two cell only i.e only one-half belongs to any one cell. • FCC cell has four atom per unit cell. • Total no. of atoms belongs to any one cell is ( x8) + ( x 6 )=4 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 13 8 1 2 1 FCC
- 14. Coordination Number and Atom positions in cubic unit cell •The number of nearest equidistant neighbors around any lattice point in the crystal lattice. •Coordination number decides whether the structure is loosely or closely packed. • The position of nearest atom can be term as follow in SCC and a is the length of each edge. The coordination of nearest neighbour at O are; ( a,0,0),( 0, a,0),( 0,0, a) 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 14 (0,a,0) (0,0,a) O (0,0,0) (1) (a,0,0)(- a,0,0) (2) (0,0,-a) (0,-a,0) (6) (4) (5) (3) axis.z,y,x,alongvectorsare,,z,, kjiwherecbjia
- 15. Coordination Number in BCC • The distance between any two nearest neighbour can be determine as follows In triangle ABC ( AC)2= (AB)2 + (BC)2 = a2 + a2 AC = a In triangle ADC ( AD)2= (AC)2 + (CD)2 = AD = a Thus the distance between any two nearest neighbour is a Similarly for FCC : The coordination number of FCC is twelve and the distance between any two nearest neighbour is a/ 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 15 a a a A B C D O a 2 3 2 a 2 2 3 2 3 2 aa 22 2
- 16. The atomic radius is half a distance between any nearest neighbors in crystal of pure element .it is necessary to calculate the atomic radius of atom of sc,bcc and fcc it should be remember that any two nearest neighbor touches each other. For SCC: If r be the atomic radius and a be the length of the edge of the cube, then AB = 2r = a or r = 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 16 a a A B 2 a SIMPLE CUBIC CELL Atomic radius
- 17. FOR BCC CUBIC CELL 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 17 4r A BC F a r 2r r C F ( AC)2= (AB)2 + (BC)2 = a2 + a2 =2 a2 ( FC)2= (AC)2 + (AF)2 = 3 a2 = 2 a2 + a2 FC = 4r r= 4 3 a The radius of each atom of a BCC cell is 4 3 a
- 18. For FCC 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 18 A B CD (a) FCC has six atoms at the center of each six faces and eight at each of the eight corner of the cube. Hence the radius of each atom of a FCC cell is B CD a 4r (b) ( BD)2= (DC)2 + (BC)2 (BD)2= (a)2 + (a)2 (4r)2 = 2(a)2 r= 2 4 a
- 19. ATOMIC PACKING FACTOR (APF) •Space occupied by all atoms in a unit cell. OR •Ratio of the volume of the atoms per unit cell to the volume of the unit cell. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 19 itcellVolumeofun celltomperunitVolumesofa ingfactorAtomicPack
- 20. SIMPLE CUBIC CELL No. of atom per unit cell = 01 Volume of one atom = 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 20 3 4 πr3 Volume of one atom of SCC lattice = (a/2 )3π = 3 4 a 6 3 SCC has one atom per unit cell Volume of atom per unit cell = No. of atoms per unit cell x volume of unit cell = 1x = Volume of SCC cell = (a)3 Atomic Packing factor = = = = 0.52 APF for SCC is 0.52 or 52% Hence SCC is loosely Packed structure. a 6 3 a 6 3
- 21. FOR BCC CUBIC CELL 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 21 No. of atom per unit cell = 02 Volume of one atom = Volume of one atom of BCC lattice = = 3 4 3 r 3 2) 4 3 (4 a a 3 3 16 Volume of per unit cell = 02x APF = 02x = = 0.68 APF of BCC is0.68 or 68% Hence BCC structure is closely packed structure. a a3 3 16 3 3 8 a a3 3 16 3
- 22. FOR FCC CUBIC CELL Number of atom per unit cell = 04 Volume of one atom of FCC lattice = = 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 22 3 4 3 r 3 4 _)2 4 ( 3 a Volume of atom per unit cell= No. of atoms per unit cell x volume of one atom = APF = = 0.74 APF for FCC is 0.74 or 74% Hence FCC is closely packed structure a 3 212 4 23 3 3 a a
- 23. Crystal structure of sodium chloride (NaCl) Or Rock Salt 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 23 a Cl Na 8 1 2 1 X 8 + x 6 = 4 Cl 12 x + 1 = 4 Na 4 1
- 24. Crystal structure of sodium chloride (NaCl) Or Rock Salt • NaCl ions is arranged in cubic pattern. • Cl ions are situated at each corner (8) and the center of each face (6) of cell. • Cl ions lie on FCC lattice. • The Na ions are also arranged in the face centered cubic lattice. • One Na ions is at center and others are located at the mid point of the twelve edges. • The position of ions in unit cell are : Na;( , , ),(0,0, ),(0, ,0), ( ,0,0 ) Cl; (0,0,0) ( , ,0),( ,0, ),( 0, , ) 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 24 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1
- 25. Unit cell and lattice constant of a space lattice • Consider a cubic crystal (a=b=c) lattice of lattice constant a and be the density of crystal material, then Volume of unit cell = mass of the unit cell = ……………….(1) Let n be the No. of atom per unit cell,M the molecular weight and N the Avogadro number then, Mass of each molecule = Mass of unit cell,m = n ………………(2) Comparing equation (1) and (2) ,we get = n = Thus lattice constant can be determined. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 25 a 3 N M N M a 3 N M a 3 N nM
- 26. 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. • Atomic packing factor = • Metallic crystals have closest packing in two forms • (i) hexagonal close packed • (ii) face- centered cubic with packing factor 74%. • The packing factor of simple cubic structure is 52%. • The packing factor of body centered cubic structure is 68%. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 26 itcellvolumeofun itcelleatomperunvolumeofth Atomic packing factor
- 27. In a crystal orientation of planes or faces can be described in terms 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. These indices are called Miller indices generally represented by (h k l). 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 27 MILLER INDICES X Y A B C Z pa qb rc O
- 28. Procedure to determine Miller Indices • Choose the plane that does not pass through the origin at (0,0,0). • Find the intercept OX,OY,OZ.let it be pa, qb&rc,where a,b,c are primitives and p,q,r may be integer or fraction. • Take reciprocal of these intercepts as follows ( , , ) Reduce the to three smallest hole number having the same ratio.the smallest possible integers h,k,l are: h:k:l = , , It is done by multiplying the reciprocal by LCM. these numbers are known as Miller indices. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 28 p 1 q 1 r 1 p 1 q 1 r 1
- 29. Reciprocal Lattice • P P Ewald devised reciprocal lattice. • Each set of parallel plane can be represented by normal to these planes having equal length equal to the reciprocal of the interplanar spacing of the corresponding set. • The normal's are drawn from a common origin and points marked at the end of the normal. • The points at the end of the normal form a lattice array. • Since the distance in the array are reciprocal to distance in the crystal, the array of points called reciprocal lattice of the crystal. • The point in the reciprocal lattice are called reciprocal lattice points. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 29
- 30. Reciprocal lattice • Real lattice 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 30
- 31. Continued……….. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 31
- 32. X – Rays Diffraction • When Wilhelm Rőntgen first discovered X-rays in 1895, no one had any about phenomenon was - hence the name X-rays (although they were referred to as Rőntgen rays for a while). • X- rays are electromagnetic radiation of exactly the same nature as light but of very much shorter wavelength between(0.01 – 10)A0. • When electrons with high value of kinetic energy collide with the target metal of high melting point and large atomic weight, x-rays are produced. In 1913, Dr Coolidge introduced new type of tube Which now most commonly used to produced x – rays. • in this tube filament is heated to produced electrons by thermionic emission and the incident on the target, after striking the target, X – rays produced. Since most KE of the electrons is absorbed by the target metal in the form of heat, a cooling system is required to maintain the temperature. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 32
- 33. Important Properties of X-rays: Properties of X-Rays • X – rays are electromagnetic radiations like light.thus they posses all the properties of electromagnetic radiations and shows a phenomena like reflection, refraction, diffraction and interference. • X-rays have very short range of wavelength,lying between (0.01 – 10)A0. • X – rays are absorbed by the material. • X – rays are not deviated by either electric or magnetic field. • X –rays ionize the gas through which they pass. • X – rays can penetrate the solid materials. • X – rays affect the photographic plate. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 33
- 34. X – Rays Production Coolidge tube • Electron beam 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 34 Heavy metal target Electron beam Filament Vaccume HT X-Ray Beam Copper Anode
- 35. Diffraction of X –rays • For testing of wave nature of X – rays, scientists tried to obtain the diffraction pattern of X – rays, man-made transmission grating (ruled with 6000 lines per inch) could not produce appreciable amount of diffraction with X – rays. thus it was concluded that diffraction of X –rays by manmade grating is impossible. Therefore to obtained the diffraction pattern of a wave the transmission grating having 40 million lines per cm are required for diffracting the X – rays. • In 1913, German physicist Max Von Laue suggested that atoms in a natural crystal are regularly arranged at equal distances, the separation between successive layers is of the order of the wavelength of X – rays so crystal act as space grating. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 35
- 36. Laue Experimental demonstration • Determination of crystal structure: 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 36
- 37. Laue Experimental demonstration: •In 1913,Laue,in collaboration with W Friedrich and P Knipping was successful in obtaining diffraction pattern of X –rays passing through the three dimensional crystal gratting.the experimental arrangement of laue demonstration is shown above. •X – rays are produced from the Coolidge tube and are collimated into fine pencil beam by passing through pin holes, the beam is allowed to pass through the crystal of ZnS or NaCl which is acting as a grating. •diffraction pattern is obtained at photographic plate. This pattern consist of a central spot, surrounded by a series of spot in a definite pattern. The symmetrical pattern of spot is called Laue's spots. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 37
- 38. Continued……… • Laue spot prove that X – rays are electromagnetic waves. • The proper explanation of laue is given by W.L Braggs he said that these spots corresponds to the constructive interference between the rays reflected from the various set of parallel crystal planes, there by satisfying the equation,2dsin = n,where is the glancing angle, d is the inter-atomic space and λ is the wavelength of the x – rays.Laue,s established following two facts: • X- Rays are electromagnetic waves of short wavelength. • In crystal, atoms are arranged in a three- dimensional lattice. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 38
- 39. Bragg reflection • The diagram below shows X-rays being reflected from a crystal. Each layer of atoms acts like a mirror and reflects X- rays strongly at an angle of reflection that equals the angle of incidence. The diagram shows reflection from successive layers 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 39 d d A B C
- 40. Bragg reflection • If the path difference between the beams from successive layers of atoms is a whole number of wavelengths, then there is constructive interference. • The path difference is the distance (AB + BC )as above: AB = BC =dsinθ (AB + BC) = 2dsinθ • The reflected beam has maximum intensity when 2dsinθ = nλ 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 40
- 41. Braggs spectrometer • W H Braggs and his son W L Braggs devised a spectrometer in which a crystal is used as a reflection grating instead of transmission grating. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 41
- 42. Continued……… • It consist of a source of X- rays ,slits and crystal mounted on the prism table consisting of two venire scale to note down the angle, is attached with the prism table. An ionization chamber is attached with the prism table, along with a galvanometer. • rays from the Coolidge tube are passed through slits to obtain pencil beam. • This beam is allowed to fall on the crystal mounted on the prism table, capable of rotating about a vertical axis passing through the center of the turn table. • the beam after reflection enter into the ionization chamber through slit the chamber is filled with ethyl bromide, and is capable of rotating with the prism table the venire scale gives the position of the ionization chamber the position of crystal and and ionization chamber is arranged in such a way that the rotation of angle in the position of crystal produces the rotation of 2 in the position of ionization chamber. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 42
- 43. Continued……… • Due to this arrangement the X- ray beam reflected from the surface of the crystal is always received in the ionization chamber. the intensity of X- rays in terms of the ionization current is observed for different values of the glancing angle. the resulting ionization current is observed through a galvanometer. • The graph plotted between the ionization current I and glancing angle for the sodium chloride crystal. • It is clear from the graph for a certain values of the ionization current I increases abruptly, the peaks of the curve for glancing angle satisfy Bragg’s equation i.e. 2dsin = nλ For n=1 λ = 2dsin 1 N=2 2λ = 2dsin 2 N=3 3λ = 2dsin 3 Hence sin1: sin2: sin3 = 1: 2 : 3 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 43
- 44. Continued……… • Thus , λ, and d if two are known then the value of the third one can be calculated. From the above experiment the following fact can be calculated. • As the order of spectrum increases, the intensity of the reflected rays decreases. • The ionization current does not falls to zero for any values of the glancing angle ,but it does not attain maximum values for certain glancing angles. it indicate that there is a continuous spectrum. 1/23/2017 Dr A K Mishra, Academic Coordinator, JIT Jahangirabad 44 2 Crystal X-Ray

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