Chapter 3 The Structure of Crystalline Solids ISSUES TO EXPLORE What is the difference in atomic arrangement between crystalline and noncrystalline solids What are the crystal structures of metals What are the characteristics of crystal structures How are crystallographic points directions and planes specified What characteristics of a material s atomic structure determine its density Chapter 3 1 Energy and Packing Non dense random packing Energy Dense ordered packing Energy typical neighbor bond length typical neighbor bond length r r typical neighbor bond energy typical neighbor bond energy Ordered structures tend to be nearer the minimum in bonding energy and are more stable Chapter 3 2 Materials and Atomic Arrangements SiO2 Crystalline materials atoms arranged in periodic 3D arrays metals many ceramics some polymers Noncrystalline materials atoms have no periodic arrangement complex structures rapid cooling Amorphous Noncrystalline Si Oxygen Fig 3 24 Callister Rethwisch 10e Chapter 3 3 Metallic Crystal Structures Atomic Packing Metals have densely packed crystal structures Why Bonds between metal atoms are nondirectional Small nearest neighbor distances to lower bond energy Free electron cloud provides high degree of shielding for ion cores Metallic crystal structures are simpler than those of ceramics and polymers Chapter 3 4 Definitions Coordination Number number of nearest neighbor or touching atoms Atomic Packing Factor APF Volume of atoms in unit cell Volume of unit cell assume hard spheres atoms unit cell x volume atom volume unit cell Chapter 3 5 Simple Cubic SC Crystal Structure Atom centers located at the eight corners of a cube ex Po Rare due to low packing density only Po has this structure Coordination 6 nearest neighbors Cube edges are close packed directions Adapted from Fig 3 3 Callister Rethwisch 10e Chapter 3 6 Atomic Packing Factor APF for Simple Cubic a R 0 5a APF 0 52 atoms unit cell 1 0 5a 3 volume atom 4 3 a3 volume unit cell close packed directions Unit cell contains 1 atom 8 x 1 8 1 atom unit cell Chapter 3 7 Body Centered Cubic Structure BCC ex Cr W Fe Ta Mo Atoms located at 8 cube corners with a single atom at the center ex Cr W Fe Ta Mo Note All atoms are identical the center atom is shaded differently for ease of viewing Coordination 8 2 atoms unit cell 1 center 8 corners x 1 8 Close packed directions run diagonally through center of unit cell Adapted from Fig 3 2 Callister Rethwisch 10e Chapter 3 8 Atomic Packing Factor BCC APF for the body centered cubic structure 0 68 4R a3 a a2 For close packed directions R 3 a 4 volume atom Chapter 3 9 R a 4 3 2 3a 4 3 a3 atoms unit cell volume unit cell APF Face Centered Cubic Structure FCC ex Al Cu Au Pb Ni Pt Ag Atoms located at 8 cube corners and at the centers of the 6 faces Note All atoms in the animation are identical Coordination 12 4 atoms unit cell 6 face x 1 2 8 corners x 1 8 Close packed directions go along face diagonals Adapted from Fig 3 1 Callister Rethwisch 10e Chapter 3 10 Atomic Packing Factor FCC APF for the face centered cubic structure 0 74 2 a a maximum achievable APF For close packed directions i e R 2a 4 4R 2 a Unit cell contains 6 x 1 2 8 x 1 8 4 atoms unit cell atoms unit cell APF 4 4 3 2a 4 3 a3 volume atom 0 74 volume unit cell Chapter 3 11 Hexagonal Close Packed Structure HCP a 2r c a Volume of atoms Volume of unit cell A 0 74 Chapter 3 12 Theoretical Density for Metals Density Mass Total of Atoms Volume in of Unit Unit Cell Cell n A VCNA where n number of atoms unit cell A atomic weight g mol VC Volume of unit cell a3 for cubic m3 NA Avogadro s number 6 022 x 1023 atoms mol Chapter 3 13 Theoretical Density Computation for Chromium Cr has BCC crystal structure A 52 00 g mol R 0 125 nm n 2 atoms unit cell a 4R 3 0 2887 nm R a atoms unit cell VC a3 2 406 x 10 23 cm3 2 52 00 g mol An VC NA volume unit cell 2 406 x 10 23 6 022 x 1023 actual 7 15 g cm3 7 19 g cm3 atoms mol Chapter 3 14 Densities Comparison for Four Material Types In general metals ceramics polymers Why Graphite Ceramics Semicond Metals Alloys Composites Polymers fibers 30 Platinum Gold W Tantalum Silver Mo Cu Ni Steels Tin Zinc Titanium Aluminum Magnesium Zirconia Al oxide Diamond Si nitride Glass soda Concrete Silicon Graphite Metals close packing metallic bonding often large atomic masses 3 m c g Ceramics Polymers often lighter elements low packing density often amorphous lighter elements C H O Composites moderate to low densities 20 10 5 4 3 2 1 0 5 0 4 0 3 Glass fibers GFRE Carbon fibers CFRE Aramid fibers AFRE PTFE Silicone PVC PET PS PE Wood Based on data in Table B1 Callister GFRE CFRE AFRE are Glass Carbon Aramid Fiber Reinforced Epoxy composites values based on 60 volume fraction of aligned fibers in an epoxy matrix Chapter 3 15 Polymorphism Allotropy Two or more distinct crystal structures for the same material Video of Tin 27 increase in volume Density decreases from 7 3 to 5 77 g cm3 Chapter 3 16 Polymorphism Allotropy Two or more distinct crystal structures for the same material Titanium or forms Carbon diamond graphite Iron system T liquid 1538 C Fe BCC 1394 C Fe FCC 912 C Fe BCC t e r u a r e p m e T Chapter 3 17 Break Chapter 3 18 Point Coordinates A point coordinate is a lattice position in a unit cell Determined as fractional multiples of a b and c unit cell edge lengths Example Unit cell upper corner 1 Lattice position is a b c 2 Divide by unit cell edge lengths a b and c and remove commas z c a b c 111 y b a x 3 Point coordinates for unit cell corner are 111 Chapter 3 19 Practice Problem Point Coordinate Practice Determine the location of the point with indices 1 a 1 b c 0 48nm 0 12 nm 1 0 46 nm 0 46 nm 1 0 40 nm 0 20 nm Chapter 3 20 Practice Problem Point Coordinate Practice Determine the point coordinates for each point in this unit cell Point Coordinates q r s 1 2 3 4 5 6 7 8 9 Chapter 3 21 Crystallographic Directions I A vector between two points z pt 2 head pt 1 tail y x ex pt 1 x1 0 y1 0 z1 0 pt 2 x2 a y2 0 z2 c 2 c 2 0 0 0 a 0 b a c Algorithm determine direction indices 1 Coordinates tail x1 y1 z1 head x2 y2 z2 2 Subtract tail from head coordinates 3 Normalize coordinate differences in terms of lattice parameters a b and c z2 …