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Chapter 1 Crystallography

Chapter 1 Crystallography. 12/19/2019. 1. 1. Outline. 1. Introduction to bonding in solids. 2. Types of bonding. 3. Classification of solids. 4. Basic definitions. 5. Crystal Systems and Bravais Lattice. 6. Miller Indices and Problems. 7. XRD Techniques 12/19/2019. 2. 2.

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Chapter 1 Crystallography

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  1. Chapter 1 Crystallography 12/19/2019 1 1

  2. Outline 1. Introduction to bonding in solids 2. Types of bonding 3. Classification of solids 4. Basic definitions 5. Crystal Systems and Bravais Lattice 6. Miller Indices and Problems 7. XRD Techniques 12/19/2019 2 2

  3. Introduction Potential energy versus interatomic distance curve 12/19/2019 3

  4. Types of Bonding 12/19/2019 4 4

  5. 5 12/19/2019

  6. 12/19/2019 6 6

  7. Basic Definitions a 12/19/2019 7 7

  8. y Translation vector B  b b  2a x    O a O B 2a b   12/19/2019 8

  9. Basis Basis Basis Lattice + Basis = Crystal structure 12/19/2019 9 9

  10. Unit Cell 12/19/2019 10 10

  11. UNIT CELL Non-primitive Primitive Simple cubic(sc) Conventional = Primitive cell Body centered cubic(bcc) Conventional ≠ Primitive cell 11 12/19/2019 11

  12. Crystallographic axes & Lattice parameters 12/19/2019 12 12

  13. Crystal systems 1. Cubic Crystal System = = = 90° a = b = c 11/28/15 12/19/2019 13 13

  14. 2.Tetragonal system = = = 90° a = b c 11/28/15 12/19/2019 14 14

  15. 3. Orthorhombic system a b c = = = 90° 11/28/15 12/19/2019 15 15

  16. 4. Monoclinic system = = 90°, 90° a b c 11/28/15 12/19/2019 16 16

  17. 5. Triclinic system a b g a b c 90° 11/28/15 12/19/2019 17 17

  18. 6. Rhombohedral (Trigonal) system = = 90° a = b = c 11/28/15 12/19/2019 18 18

  19. 7. Hexagonal system a = b c = = 90°, = 120° 11/28/15 12/19/2019 19 19

  20. Bravais Lattice An infinite array of discrete points generated by a set of discrete translation operations described by Rn a n a n a  11 22 33 where niare integers and aiare primitive vectors 14 Bravais lattices are possible in 3- dimentional space. 11/28/15 12/19/2019 20 20

  21. Lattice types Base Face Primitive (P) Body centered (C) centered (F) centered (I) 1 8 1 8 n     1 8 1 2 1 8 1 2 n       2       n       6       8 1 n   1 2 8   8     2 8     4 11/28/15 12/19/2019 21 21 Lattice types (animation)

  22. 12/19/2019 22

  23. 11/28/15 12/19/2019 23 23

  24. Relation between atomic radius and edge length 11/28/15 12/19/2019 24 24

  25. Face centered cubic Body centered cubic Simple cubic Z = 4 /2 2 r a  Z = 2 3 /4 a r  Z = 1 /2 r a  4 3 4 3 4 3   3 3 r r  3 r P F = Z P F = Z   P F = Z  3 3 a a 3 a PF = 52% 11/28/15 12/19/2019 PF = 74% PF = 68% 25 25

  26. Different lattice planes in a crystal d 11/28/15 12/19/2019 26 26

  27. Crystal planes 11/28/15 12/19/2019 27 27

  28. Inter-planar spacing in Crystals 11/28/15 12/19/2019 28 28

  29. Inter-planar spacing in different crystal systems 11/28/15 12/19/2019 29 29

  30. Problems on Miller indices Q1: Q2: 11/28/15 12/19/2019 30 30

  31. Q3: Determine the miller indices for the planes shown in the following unit cell 11/28/15 12/19/2019 31 31

  32. Q4: What are Miller Indices? Draw (111) and (110) planes in a cubic lattice. Q5: Sketch the following planes of a cubic unit cell (001), (120), (211) Q6: Obtain the Miller indices of a plane which intercepts at a, b/2 and 3c in simple cubic unit cell. Draw a neat diagram 11/28/15 12/19/2019 32 32

  33. Problems on inter-planar spacing 1.Explain how the X-ray diffraction can be employed to determine the crystal structure. Give the ratio of inter-planar distances of (100), (110) and (111) planes for a simple cubic structure. 11/28/15 12/19/2019 33 33

  34. 2. The distance between (110) planes in a body centered cubic structure is 0.203 nm. What is the size of the unit cell? What is the radius of the atom? 12/19/2019 34

  35. Reciprocal lattice 12/19/2019 35

  36. Bragg’s Law: n = 2dsin 11/28/15 12/19/2019 36

  37. 11/28/15 12/19/2019 37

  38. Problem on Bragg’s law 1. 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 diffraction. the second-order 12/19/2019 38 38 11/28/15

  39. Problem on Bragg’s law 2. A beam of X-rays is incident on a NaCl crystal with lattice plane spacing 0.282 nm. Calculate the wavelength of X-rays if the first-order Bragg reflection takes place at a glancing angle of 8 °35′. Also calculate the maximum order of diffraction possible. 12/19/2019 39

  40. Problem on Bragg’s law 3. Monochromatic X-rays of λ = 1.5 A.U are incident on a crystal face having an inter- planar spacing of 1.6 A.U. Find the highest order for which Bragg’s reflection maximum can be seen. 12/19/2019 40

  41. Problem on Bragg’s law 4.For BCC iron, compute (a) the inter- planar spacing, and (b) the diffraction angle for the (220) set of planes. The lattice parameter for Fe is 0.2866 nm. Also, assume that monochromatic having a wavelength of 0.1790 nm is used, and the order of reflection is 1. radiation 12/19/2019 41

  42. Problem on Bragg’s law 5.The metal niobium has a BCC crystal structure. If the angle of diffraction for the (211) set of planes occurs at 75.99o (first order reflection) when monochromatic X- radiation having a wavelength 0.1659 nm is used. Compute (a) the inter-planar spacing for this set of planes and (b) the atomic radius for the niobium atom. 12/19/2019 42

  43. Laue Method D CollimatorSingle F crystal 2 r1 X-rays B S D r D   tan2 Transmission Method 1 11/28/15 12/19/2019 43 43

  44. Back-reflection method Collimator F Single crystal (180 -2 ) r2 B S D t a n ( 1 8 02)r   2 D 12/19/2019 44

  45. Transmission method Back-reflection method 11/28/15 12/19/2019 45 45

  46. Rotating Crystal Method Experimental setup of Rotation Crystal method Single crystal Cylindrical film Axis of Crystal 46 11/28/15 12/19/2019 46

  47. 11/28/15 12/19/2019 47 47

  48. 11/28/15 12/19/2019 48 48

  49. Applications of XRD Determination of Lattice parameter  S W  S W 1          2   1 2 2  2 2 2 2 a     2  2 s i n4hk l 11/28/15 12/19/2019 49 49

  50. X-Ray Diffraction technique is used to Distinguishing between crystalline & amorphous materials. Determination of crystalline materials. Determination of electron distribution within the atoms, & throughout the unit cell. the structure of 11/28/15 12/19/2019 Confidential 50 50

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