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- 1. Crystal Structure and X ray Diffraction Unit I Dr Md Kaleem Department of Applied Sciences Jahangirabad Institute of Technology (JIT), Jahangirabad, Barabanki(UP) - 225203 1/31/2017 1DR MD KALEEM/ ASSISTANT PROFESSOR
- 2. • Relationship between structures of engineering materials • To understand the classification of crystals • To understand mathematical description of ideal crystal • To understand Miller indices for directions and planes in lattices and crystals • To understand how to use X-Ray Diffraction for determination of crystal geometry 1/31/2017 2DR MD KALEEM/ ASSISTANT PROFESSOR
- 3. 1/31/2017 3DR MD KALEEM/ ASSISTANT PROFESSOR
- 4. Solid can be divided in two categories on the basis of periodicity of constituent atoms or group of atoms • Crystalline solids consists of atoms, ions or molecules arranged in ordered repetitive array e.g: Common inorganic materials are crystalline – Metals : Cu, Zn, Fe, Cu-Zn alloys – Semiconductors: Si, Ge, GaAs – Ceramics: Alumina (Al2O3), Zirconia (Zr2O3), SiC, SrTiO3. • Non crystalline or Amorphous consists of atoms, ions or molecules arranged in random order e.g: organic things like glass, wood, paper, bone, sand; concrete walls, etc Crystalline Solids grains crystals 1/31/2017 4DR MD KALEEM/ ASSISTANT PROFESSOR
- 5. 1/31/2017 5DR MD KALEEM/ ASSISTANT PROFESSOR Crystal = Lattice + Motif Lattice : regular repeated three-dimensional arrangement of points Motif/ Basis: an entity (typically an atom or a group of atoms) associated with each lattice point
- 6. 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 6 Lattice where to repeat Motif what to repeat Lattice: Translationally periodic arrangement of points Crystal: Translationally periodic arrangement of motifs
- 7. Space lattice: An array of points in space such that every point has identical surroundings Unit Cell: It is basic structural unit of crystal, with an atomic arrangement which when repeated three dimensionally gives the total structure of the crystal Lattice Parameters: It defines shape and size of the unit cell Three lattice vector (a, b, c) and interfacial angle (, , ) are known as lattice parameters 1/31/2017 7DR MD KALEEM/ ASSISTANT PROFESSOR
- 8. Unit cell with lattice points at the corners only, called primitive cell. Unit cell may be primitive cell but all primitive cells are not essentially unit cells. 1/31/2017 8DR MD KALEEM/ ASSISTANT PROFESSOR
- 9. • Crystallographers classified the unit cells into seven possible distinct types of unit cells by assigning specific values to lattice vector (a, b, c) and interfacial angle (, , ) called seven crystal system. 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 9
- 10. Crystal System Lattice Vector Interfacial Angle Example 1 Cubic a = b = c = = = 90o NaCl, CaF2, Au, Ag, Cu, Fe 3 Tetragonal a = b ≠ c = = = 90o TiO2, NiSO4, SnO2 3 Orthorhombic a ≠ b ≠ c = = = 90o KNO3, BaSO4, PbCO3, Ga 4 Monoclinic a ≠ b ≠ c = = 90o≠ CaSO4.2H2O (Gypsum), FeSO4 5 Triclinic a ≠ b ≠ c ≠ ≠ ≠ 90o CuSO4, K2Cr2O7 6 Trigonal a = b = c = = ≠ 90o As, Sb, Bi, Calcite 7 Hexagonal a = b ≠ c = = 90o, =120o SiO2, AgI, Ni, As, Zn, Mg 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 10
- 11. 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 11
- 12. • A. J. Bravais in 1948 shown that with the centering (face, base and body centering) added to these, 14 kinds of 3D lattices, known as Bravais lattices. 1/31/2017 12DR MD KALEEM/ ASSISTANT PROFESSOR
- 13. Coordination Number: It is defined as the number of nearest neighbors around any lattice point in the crystal lattice. 1/31/2017 13DR MD KALEEM/ ASSISTANT PROFESSOR
- 14. •Miller indices for crystallographic planes •Miller notation system (hkl) •Miller index – the reciprocals of the fractional intercepts that the plane makes with the x, y, and z axes of the three nonparallel edges of the cubic unit cell William Hallowes Miller 1/31/2017 14DR MD KALEEM/ ASSISTANT PROFESSOR
- 15. • Choose a plane not pass through (0, 0, 0) • Determine the intercepts of the plane with x, y, and z axes • Form the reciprocals of these intercepts • Find the smallest set of whole numbers that are in the same ratio as the intercepts 1/31/2017 15DR MD KALEEM/ ASSISTANT PROFESSOR
- 16. • Find the Miller Indices of the plane which cuts off intercepts in the ratio 1 a:3b:-2c along the three co-ordinate axes, where a, b and c are the primitives. • If pa, qb and rc are the intercepts of the given set of planes on X-, Y-, and Z- axes respectively then, pa: qb: rc= 1 a:3b:-2c or p:q:r=1:3:-2 so 1/p : 1/q : 1/r = 1/1 :1/3 : -1/2 LCM of 1, 3 and 2 = 6, so multiply by it 1/p : 1/q : 1/r = 6:2:-3 Thus the Miller Indices of the plane is (6 2 ) 1/31/2017 16DR MD KALEEM/ ASSISTANT PROFESSOR 3
- 17. 1/31/2017 17DR MD KALEEM/ ASSISTANT PROFESSOR
- 18. 1/31/2017 18DR MD KALEEM/ ASSISTANT PROFESSOR
- 19. 1/31/2017 19DR MD KALEEM/ ASSISTANT PROFESSOR
- 20. 1/31/2017 20DR MD KALEEM/ ASSISTANT PROFESSOR
- 21. • It is infinite periodic three dimensional array of reciprocal lattice points whose spacing varies inversely as the distances between the planes in the direct lattice of the crystal. 1/31/2017 21DR MD KALEEM/ ASSISTANT PROFESSOR
- 22. Take some point as an origin From this origin, lay out the normal to every family of parallel planes in the direct lattice; Set the length of each normal equal to 2p times the reciprocal of the interplanar spacing for its particular set of planes; Place a point at the end of each normal. 1/31/2017 22DR MD KALEEM/ ASSISTANT PROFESSOR
- 23. • Any diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal whereas the microscopic image is a map of the direct lattice. • While the primitive vectors of a direct lattice have the dimensions of length those of the reciprocal lattice have the dimensions of (length)− 1. • Direct lattice or crystal lattice is a lattice in ordinary space or real space. Reciprocal lattice is in reciprocal space or k-space or Fourier space. • The direct lattice is the reciprocal of its own reciprocal lattice. • The reciprocal lattice of a simple cubic lattice is also a simple cubic lattice. • The reciprocal lattice of a face centered cubic lattice is a body centered cubic lattice. • The reciprocal lattice of a body centered cubic lattice is a face centered cubic lattice, and 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 23
- 24. NaCl has a cubic unit cell. It is best thought of as a face- centered cubic array of anions with an interpenetrating fcc cation lattice (or vice-versa) The cell looks the same whether we start with anions or cations on the corners. Each ion is 6-coordinate and has a local octahedral geometry. 1/31/2017 24DR MD KALEEM/ ASSISTANT PROFESSOR
- 25. • The Bravais space lattice of NaCl is truly fcc with a basis of one Na+ ion one Cl- ion separated by one half the body diagonal (a√3/2) of a unit cube. • There are four pair of Na+ and Cl- ions present per unit cell. • The position of ions in unit cell are • Na+ : (½, ½, ½), (0,0, ½), (0, ½,0), (½,0,0) • Cl- : (0,0,0), (½, ½,0), (½,0, ½), (0, ½, ½) 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 25
- 26. For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength In crystals the typical inter-atomic spacing ~ 2-3 Å so the suitable radiation is X-rays Hence, X-rays can be used for the study of crystal structures 1/31/2017 26DR MD KALEEM/ ASSISTANT PROFESSOR
- 27. The path difference between rays = 2d Sin For constructive interference: n = 2d Sin 1/31/2017 27DR MD KALEEM/ ASSISTANT PROFESSOR
- 28. • Q. A beam of X-rays of wavelength 0.071 nm is diffracted by (110) plane of rock salt with lattice constant of 0.28 nm. Find the glancing angle for the second-order diffraction. • Given data are: • Wavelength (λ) of X-rays = 0.071 nm, Lattice constant (a) = 0.28 nm Plane (hkl) = (110), Order of diffraction = 2 Glancing angle θ = ? Bragg’s law is 2d sin θ = nλ 1/31/2017 28DR MD KALEEM/ ASSISTANT PROFESSOR
- 29. Substitute in Bragg’s equation 1/31/2017 29DR MD KALEEM/ ASSISTANT PROFESSOR
- 30. Bragg’s spectrometer method is one of the important method for studying crystals using X-rays. The apparatus consists of a X-ray tube from which a narrow beam of X- rays is allowed to fall on the crystal mounted on a rotating table. The rotating table is provided with scale and vernier, from which the angle of incidence, θ can be measured. 1/31/2017 30DR MD KALEEM/ ASSISTANT PROFESSOR
- 31. • Bragg’s spectrometer is used to determine the structure of crystal. • The ratio of lattice spacing for various groups of planes are obtained by using Bragg’s Law. • The ratio would be different for different crystals • By comparing those known standard ratios with experimentally determined ratios, crystal structure can be obtained. 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 31
- 32. • If for a particular crystal having interplaner spacing d1, d2, d3 strong Bragg’s reflection occur at glancing angle θ1, θ2, θ3 then from Bragg’s law • 2d1sin θ1=λ, 2d2sin θ2=λ, 2d3sin θ3=λ • So, d1: d2: d3 = 1/sin θ1= 1/sin θ2=1/sin θ3 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 32
- 33. • For KCl Crystal, Bragg’s obtained strong Bragg’s reflection at θ1= 5o23’, θ2=7o37’, θ3=9o25’’for planes (100), (110) and (111) • So, d100: d110: d111= 1/sin 5o23’= 1/sin 7o37’=1/sin 9o25’ = 1:1/√2:1/√3 • This corresponds to theoretical result for simple cubic lattice . Therefore it is concluded that KCl crystal has simple cubic structure. 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 33
- 34. • When light encounters charged particles, the particle interact with light and cause some of the light to be scattered. This is called Compton Scattering. 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 34
- 35. • Arthur H. Compton in 1923 observed that when electromagnetic wave of short wavelength (X ray) strikes an electron, an increase in wavelength of X-rays or gamma rays occurs when they are scattered. 1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 35 cos1 cm h e if

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