Chapter 1 Crystallography 12/6/2018 1 1 1

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 12/6/2018 2 2

Potential energy versus interatomic distance curve Introduction Potential energy versus interatomic distance curve 12/6/2018 3

Types of Bonding 12/6/2018 4 4 4

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Basic Definitions a 12/6/2018 7 7

y Translation vector B 2 b x O a 12/6/2018 8

Lattice + Basis = Crystal structure 12/6/2018 9 9

Unit Cell 12/6/2018 10 10

UNIT CELL Primitive Non-primitive Simple cubic(sc) Conventional = Primitive cell Body centered cubic(bcc) Conventional ≠ Primitive cell 12/6/2018 11 11

Crystallographic axes & Lattice parameters 12/6/2018 12 12

Crystal systems 1. Cubic Crystal System a = b = c  =  =  = 90° 11/28/15 12/6/2018 13 13 13

2.Tetragonal system a = b  c  =  =  = 90° 12/6/2018 11/28/15 14 14

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

4. Monoclinic system a  b  c  =  = 90°,   90°  12/6/2018 11/28/15 12/6/2018 16 16 16

5. Triclinic system a  b  c       90° a b g 12/6/2018 11/28/15 17 17 17

6. Rhombohedral (Trigonal) system a = b = c  =  =   90° 11/28/15 12/6/2018 18 18 18

7. Hexagonal system a = b  c =  = 90°,  = 120° 12/6/2018 11/28/15 19 19 19

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. 12/6/2018 11/28/15 20 20

Lattice types (animation) Primitive (P) Body centered (I) Face centered (F) Base centered (C) 12/6/2018 11/28/15 21 Lattice types (animation) 21

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Relation between atomic radius and edge length 11/28/15 12/6/2018 24 24

Simple cubic Face centered cubic Body centered cubic Z = 4 Z = 2 Z = 1 PF = 52% PF = 74% PF = 68% 12/6/2018 11/28/15 25 25

Different lattice planes in a crystal 11/28/15 12/6/2018 26 26

Crystal planes 11/28/15 12/6/2018 27 27

Inter-planar spacing in Crystals 12/6/2018 11/28/15 28 28

Inter-planar spacing in different crystal systems 12/6/2018 11/28/15 29 29

Problems on Miller indices Q1: Q2: 11/28/15 12/6/2018 30 30

Q3: Determine the miller indices for the planes shown in the following unit cell 12/6/2018 11/28/15 31 31

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 11/28/15 12/6/2018 32 32

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. 11/28/15 12/6/2018 33 33

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? 12/6/2018 34

Reciprocal lattice 12/6/2018

Bragg’s Law: n = 2dsin 12/6/2018 11/28/15 36

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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 the second-order diffraction. 12/6/2018 38 11/28/15 38

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. 12/6/2018 39

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. 12/6/2018 40

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 radiation having a wavelength of 0.1790 nm is used, and the order of reflection is 1. 12/6/2018 41

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. 12/6/2018 42

Laue Method Collimator X-rays Method Single crystal r1 D F 2 B S D Transmission Method 11/28/15 12/6/2018 43 43 43

Back-reflection method Collimator Single crystal (180 -2 ) r2 B S D 12/6/2018 44

Back-reflection method Transmission method Back-reflection method 11/28/15 12/6/2018 45 45

Rotating Crystal Method Experimental setup of Rotation Crystal method Single crystal Cylindrical film X-rays Axis of Crystal 11/28/15 12/6/2018 46 46

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Determination of Lattice parameter Applications of XRD Determination of Lattice parameter 11/28/15 12/6/2018 49 49

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. 11/28/15 12/6/2018 Confidential 50 50 50

Determination of the orientation of single crystals. Determination of the texture of polygrained materials. Measurement of strain and small grain size…..etc. 12/6/2018 51

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. 11/28/15 12/6/2018 Confidential 52 52

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. 12/6/2018 53

Problems Chromium has BCC structure. Its atomic radius is 0.1249 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. 11/28/15 12/6/2018 Confidential 54 54

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 55.85 and 7860 kg/m3 respectively. 4. If the edge of the unit cell of a cube in the diamond structure is 0.356 nm, calculate the number of atoms/m3. 12/6/2018 55

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. 12/6/2018 56

7. A beam of X-rays of wavelength 0 7. A beam of X-rays of wavelength 0.071 nm is diffracted by (100) plane of rock salt with lattice constant of 0.28nm. Find the glancing angle for the second – order diffraction. 11/28/15 12/6/2018 Confidential 57 57

8. 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 8o 35’. Also calculate the maximum order of diffraction possible. 9. The fraction of vacant sites in a metal is 1 X 10-10 at 500oC. What will be the fraction of vacancy sites at 1000o C? 12/6/2018 11/28/15 Confidential 58 58

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 1.54 Ao are employed in a diffraction experiment. Determine the lattice parameter of nickel. 12/6/2018 59

12. Monochromatic X-rays of  = 1. 5 A 12. Monochromatic X-rays of  = 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. 13. Copper has FCC structure with lattice constant 0.36nm. Calculate the inter-planar spacing for (111) and (321) planes. 12/6/2018 60

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 0.203 nm. What is the size of the unit cell? What is the radius of the atom? 15. Monochromatic X-rays of  = 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. 12/6/2018 61

16. The first order diffraction occurs when a X-ray beam of wavelength 0.675 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. 12/6/2018 62