Atomic and Ionic Arrangements - MECN2010A PDF

Chapter 3: Atomic and Ionic Arrangements MECN2010A – Introduction to Materials Science & Engineering Chapter 3b Atomic and Ionic Arrangements Dr...

Chapter 3: Atomic and Ionic Arrangements MECN2010A – Introduction to Materials Science & Engineering Chapter 3b Atomic and Ionic Arrangements Dr Michael Lucas Email: [email protected] North West Engineering Building First floor, Office 27 ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) 1 Chapter 3: Atomic and Ionic Arrangements Atomic and Ionic Arrangements What will be addressed: How do atoms and ions assemble into solid structures. How to describe the arrangements in a crystalline solid. How the density and other properties of materials depend on its structure. How do the crystal structures of ceramic materials differ from those of metals ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) 2 Chapter 3: Atomic and Ionic Arrangements Crystals as Building Blocks Some engineering applications require single crystals: -- diamond single -- turbine blades crystals for abrasives Properties of crystalline materials often related to the crystal structure. -- Ex: Quartz fractures more easily along Some crystal planes than others. ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Single vs Polycrystals E (diagonal) = 273 GPa Single Crystals: Properties vary with direction: Anisotropic Example: the modulus of elasticity (E) in BCC iron. E (edge) = 125 GPa 200 mm Polycrystals: Properties may/may not vary with direction. If grains are randomly oriented: isotropic (Epoly iron= 210 GPa) If grains are textured. anisotropic ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Point Coordinates z 111 Point coordinates for unit cell center are c a/2, b/2, c/2 ½½½ 000 y a b Point coordinates for unit cell corner are 111 x  z 2c  Translation: integer multiple of lattice constants  identical position in another unit cell   b y b ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements HCP Points Coordinates z a2 a -a3 2 2 a3 a2 a 1 2 a3 a1 a1 dashed red lines indicate ex: ½, ½, -1, 0 projections onto a1 and a2 axes ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Crystallographic Directions – Miller Indices z Algorithm 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c y 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas x [uvw] ex: 1, 0, ½ => 2, 0, 1 => [ 201 ] -1, 1, 1 => [ 111 ] where overbar represents a negative index families of directions Ex. Different mechanical properties in different ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements HCP Crystallographic Directions z Algorithm 1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit a2 cell dimensions a1, a2, a3, or c 3. Adjust to smallest integer values - 4. Enclose in square brackets, no commas a3 [hkil] a2 a1 a2 -a3 2 ex: ½, ½, -1, 0 => [ 1120 ] a3 a1 2 dashed red lines indicate projections onto a1 and a2 axes a1 ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements HCP Crystallographic Directions 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') as follows. z [ u 'v ' w ' ] [ uvtw ] 1 u = (2 u ' - v ') 3 1 a2 v = ( 2 v ' - u ') 3 - t = - (u +v ) a3 w = w' a1 ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Linear Density # of repeat distances Linear Density of Atoms  LD = Unit length of direction vector ex: linear density of Al in direction a = 0.405 nm # repeat d. a 2 LD = = 3.5 nm -1 length 2a Linear Packing Fraction of a Direction (fraction covered by atoms) = LD * 2R ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Crystallographic Planes 11 ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Crystallographic Planes Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions. All parallel planes have same Miller indices. Algorithm move the plane out of the origin 1. Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Clear fractions 4. Enclose in parentheses, no commas i.e., (hkl) 5. Negative numbers must be written with a bar over the number ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Crystallographic Planes z example a b c 1. Intercepts 1 1  c 2. Reciprocals 1/1 1/1 1/ 1 1 0 3. Reduction 1 1 0 y a b 4. Miller Indices (110) x z example a b c 1. Intercepts 1/2   c 2. Reciprocals 1/½ 1/ 1/ 2 0 0 3. Reduction 2 0 0 y 4. Miller Indices (200) a b x * DO NOT REDUCE TO LOWEST INTEGERS ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Crystallographic Planes We want to examine the atomic packing of crystallographic planes. Iron: The atomic packing of the planes is important!! a) Draw a (100) crystallographic plane for Fe at standard room temperature. b) Calculate the planar density for this plane. ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Planar Density of (100) Iron At T < 912C iron has the BCC structure. 2D repeat unit (100) 4 3 a= R 3 atoms Radius of iron R = 0.1241 nm 2D repeat unit 1 1 atoms atoms Planar Density = = 2 = 12.1 = 1.2 x 1019 area a2 4 3 nm 2 m2 R 2D repeat unit 3 Packing Fraction: PD ( π R2 ) = 0.79 ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Hexagonal Close-Packed Planes ABAB... Stacking Sequence 2D Projection 3D Arrangement Top layer A sites Middle layer Bottom layer B sites HCP Unit Cell Coordination # = 12 A sites APF = 0.74 2 atoms/unit cell CP Planes = (0001) & (0002) ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Hexagonal Close-Packed Planes ABCABC... Stacking Sequence 2D Projection B B C A A sites B B B C C B sites B B C sites A 3D Arrangement FCC Unit Cell B C Coordination # = 12 CP Planes {111} APF = 0.74 4 atoms/unit cell ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Hexagonal Close-Packed Planes To build up the HCP: 1) start with a hexagonal close-packed array of atoms in a plane 2) place another layer of similarly arranged atoms on top so that they nestle into the valleys between the atoms in the first layer. 3) then place a third layer on top, in line with the first. If the same stacking of hexagonal close-packed layers is performed: 3b) the third layer goes into the third possible position instead of above the first the surprising result is an FCC structure. ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) 18 Chapter 3: Atomic and Ionic Arrangements Announcements Homework Examples: Problems (Ed. 3/ Ed. 2): 3-6 / 3-6 3-23 / 3-19 3-7 / 3-7 3-25 / 3-21 3-8 / 3-8 3-27 / 3-23 3-9 / 3-9 3-35 / 3-31 3-36 / 3-32 ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ ____________________________________________________ MECN2010A: Introduction to Materials Science and Engineering (2025) Chapter 3: Atomic and Ionic Arrangements Copyright Notice Printed and electronic course packs and notes distributed via Ulwazi (Canvas) are compiled to facilitate access to specific course material for registered students at the University of the Witwatersrand ("Wits") only. 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