1. Chapter 1 Crystallography
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2. Outline Introduction to bonding in solids Types of bonding
Classification of solids
Basic definitions
Crystal Systems and Bravais Lattice
Miller Indices and Problems
XRD Techniques 2
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3. Potential energy versus interatomic distance curve
Introduction
Potential energy versus interatomic distance curve
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4. Types of Bonding
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7. Basic Definitions a
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8. y
Translation vector B 2 b x O a
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9. Lattice + Basis = Crystal structure
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10. Unit Cell
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11. UNIT CELL Primitive Non-primitive
Simple cubic(sc)
Conventional = Primitive cell
Body centered cubic(bcc)
Conventional ≠ Primitive cell
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12. Crystallographic axes & Lattice parameters
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13. Crystal systems 1. Cubic Crystal System a = b = c  =  =  = 90°
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14. 2.Tetragonal system a = b  c  =  =  = 90° 12/6/2018 11/28/15 14 14

15. 3. Orthorhombic system a  b  c  =  =  = 90° 12/6/2018 11/28/15 15

16. 4. Monoclinic system a  b  c  =  = 90°,   90°  12/6/2018
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17. 5. Triclinic system a  b  c       90° a b g 12/6/2018 11/28/1517 17 17

18. 6. Rhombohedral (Trigonal) system
a = b = c
 =  =   90°
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19. 7. Hexagonal system a = b  c =  = 90°,  = 120° 12/6/2018 11/28/1519 19 19

20. where ni are integers and ai are primitive vectors
Bravais Lattice
An infinite array of discrete points generated by a set of discrete translation operations described by
where ni are integers and ai are primitive vectors
14 Bravais lattices are possible in 3- dimentional space.
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21. Lattice types (animation)
Primitive
(P)
Body centered (I)
Face centered (F)
Base centered (C)
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Lattice types (animation) 21

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24. Relation between atomic radius and edge length
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25. Simple cubic Face centered cubic Body centered cubic
Z = 4
Z = 2
Z = 1
PF = 52%
PF = 74%
PF = 68%
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26. Different lattice planes in a crystal
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27. Crystal planes
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28. Inter-planar spacing in Crystals
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29. Inter-planar spacing in different crystal systems
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30. Problems on Miller indices
Q1:
Q2:
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31. Q3: Determine the miller indices for the planes shown in the following unit cell
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32. Q4: What are Miller Indices
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 showing the plane
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33. Problems on inter-planar spacing
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.
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34. 2. The distance between (110) planes in a body centered cubic structure is nm. What is the size of the unit cell? What is the radius of the atom?
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35. Reciprocal lattice
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36. Bragg’s Law: n = 2dsin
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38. Problem on Bragg’s law
1. A beam of X-rays of wavelength 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.
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39. Problem on Bragg’s law
2. A beam of X-rays is incident on a NaCl crystal with lattice plane spacing 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.
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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.
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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 nm. Also, assume that monochromatic radiation having a wavelength of nm is used, and the order of reflection is 1.
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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 nm is used. Compute (a) the inter-planar spacing for this set of planes and (b) the atomic radius for the niobium atom.
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43. Laue Method Collimator X-rays Method Single crystal r1 D F 2 B S D
Transmission
Method
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44. Back-reflection method Collimator
Single crystal
(180 -2 ) r2 B S D
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45. Back-reflection method
Transmission method
Back-reflection method
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46. Rotating Crystal Method
Experimental setup of Rotation Crystal method
Single crystal
Cylindrical film
X-rays
Axis of Crystal
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49. Determination of Lattice parameter
Applications of XRD
Determination of Lattice parameter
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50. X-Ray Diffraction technique is used to
Distinguishing between crystalline & amorphous materials.
Determination of the structure of crystalline materials.
Determination of electron distribution within the atoms, & throughout the unit cell.
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51. Determination of the orientation of single crystals.
Determination of the texture of polygrained materials.
Measurement of strain and small grain size…..etc.
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52. Advantages & Disadvantages of
X-Ray Diffraction
Advantages
XRD is a nondestructive technique.
X-Rays are the least expensive, the most convenient & the most widely used method to determine crystal structures.
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53. X-Rays do not interact very strongly with lighter elements.
X-Rays are not absorbed very much by air, so the sample need not be in an evacuated chamber.
Disadvantages
X-Rays do not interact very strongly with lighter elements.
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54. Problems
Chromium has BCC structure. Its atomic radius is nm . Calculate the free volume / unit cell.
Lithium crystallizes in BCC structure. Calculate the lattice constant, given that the atomic weight and density for lithium are 6.94 and 530 kg/m3 respectively.
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55. 3. Iron crystallizes in BCC structure
3. Iron crystallizes in BCC structure. Calculate the lattice constant, given that the atomic weight and density of iron are and 7860 kg/m3 respectively.
4. If the edge of the unit cell of a cube in the diamond structure is nm, calculate the number of atoms/m3.
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56. 5. A metal in BCC structure has a lattice constant 3. 5Ao
5. A metal in BCC structure has a lattice constant 3.5Ao. Calculate the number of atoms per sq. mm area in the (200) plane.
6. Germanium crystallizes in diamond (from) structures with 8 atoms per unit cell. If the lattice constant is 5.62 Ao, calculate its density.
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57. 7. A beam of X-rays of wavelength 0
7. A beam of X-rays of wavelength nm is diffracted by (100) plane of rock salt with lattice constant of 0.28nm. Find the glancing angle for the second – order diffraction.
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58. 8. A beam of X-rays is incident on a NaCl crystal with lattice plane spacing nm. Calculate the wavelength of X-rays if the first-order Bragg reflection takes place at a glancing angle of 8o 35’. Also calculate the maximum order of diffraction possible.
9. The fraction of vacant sites in a metal is 1 X at 500oC. What will be the fraction of vacancy sites at 1000o C?
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59. 10. Calculate the ratios of d100 : d110 : d111 for a simple cubic structure
11. The Bragg’s angle in the first order for (220) reflection from nickel (FCC) is 38.2o. When X-rays of wavelength Ao are employed in a diffraction experiment. Determine the lattice parameter of nickel.
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60. 12. Monochromatic X-rays of  = 1. 5 A
12. Monochromatic X-rays of  = 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.
13. Copper has FCC structure with lattice constant 0.36nm. Calculate the inter-planar spacing for (111) and (321) planes.
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61. 14. The distance between (100) planes in a BCC structure is 0. 203 nm
14.The distance between (100) planes in a BCC structure is nm. What is the size of the unit cell? What is the radius of the atom?
15. Monochromatic X-rays of  = 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.
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62. 16. The first order diffraction occurs when a X-ray beam of wavelength Ao incident at a glancing angle 5o 25’ on a crystal. What is the glancing angle for third-order diffraction to occur?
17. The Bragg’s angle in the first order for (220) reflection from nickel ( FCC) is 38.2o . When X-rays of wavelength 1.54 Ao are employed in a diffraction experiment. Determine the lattice parameter of nickel.
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