Chapter 3: The Structure of Crystalline Solids

1. Chapter 3: The Structure of Crystalline Solids ISSUES TO ADDRESS... • How do atoms assemble into solid structures? • How does the density of a material depend on its structure? • When do material properties vary with the sample (i.e., part) orientation?

2. The “unit cell” is the basic repeating unit of the arrangement of atoms, ions or molecules in a crystalline solid. The “lattice” refers to the 3-D array of particles in a crystalline solid. One type of atom occupies a “lattice point” in the array.

3. Examples of Unit Cells

4. SPACE LATTICE AND UNIT CELLSSpace Lattice atoms arranged in a pattern that repeats itself in three dimensions.Unit cell smallest grouping which can be translated in three dimensions to recreate the space lattice.

5. Crystalline Versus Amorphous Quartz Obsidian

6. Crystals

7. Crystal Structures Atoms (and later ions) will be viewed as hard spheres. In the case of pure metals, the packing pattern often provides the greatest spatial efficiency (closest packing). Ionic crystals can often be viewed as a close-packed arrangement of the larger ion, with the smaller ion placed in the “holes” of the structure.

8. Unit Cells Crystals consist of repeating asymmetric units which may be atoms, ions or molecules. The space lattice is the pattern formed by the points that represent these repeating structural units.

9. Unit Cells A unit cell of the crystal is an imaginary parallel-sided region from which the entire crystal can be built up. Usually the smallest unit cell which exhibits the greatest symmetry is chosen. If repeated (translated) in 3 dimensions, the entire crystal is recreated.

10. Crystal Structures • Types of crystal structures • Face centered cubic (FCC) • Body centered cubic (BCC) • Hexagonal close packed (HCP)

11. 3 2 1 1 4 2 1

12. FCC HCP BCC

13. Face Centered Cubic (FCC) • Atoms are arranged at the corners and center of each cube face of the cell. • Atoms are assumed to touch along face diagonals

14. Face Centered Cubic (FCC) • The lattice parameter, a, is related to the radius of the atom in the cell through: • Coordination number: the number of nearest neighbors to any atom. For FCC systems, the coordination number is 12.

15. Face Centered Cubic (FCC) • Atomic Packing Factor: the ratio of atomic sphere volume to unit cell volume, assuming a hard sphere model. • FCC systems have an APF of 0.74, the maximum packing for a system in which all spheres have equal diameter.

16. Body Centered Cubic • Atoms are arranged at the corners of the cube with another atom at the cube center.

17. ATOMIC PACKING FACTOR: BCC Adapted from Fig. 3.2, Callister 6e. • APF for a body-centered cubic structure = p3/8 = 0.68

18. Body Centered Cubic • Since atoms are assumed to touch along the cube diagonal in BCC, the lattice parameter is related to atomic radius through:

19. Principal Metallic Crystal StructuresWe will concentrate on three of the more densely packed crystal structures, BCC - body centered cubic, FCC - face centered cubic, and HCP - hexagonal close packed. BCC - 2 atoms perunit cell CN = 8 

20. FCC - 4 atoms per unit cell CN = 12  HCP - 6 atoms per unit cell CN = 12

21. Hexagonal Close Packed • Cell of an HCP lattice is visualized as a top and bottom plane of 7 atoms, forming a regular hexagon around a central atom. In between these planes is a half-hexagon of 3 atoms.

22. Close Packing Since metal atoms and ions lack directional bonding, they will often pack with greatest efficiency. In close or closest packing, each metal atom has 12 nearest neighbors. The number of nearest neighbors is called the coordination number. Six atoms surround an atom in the same plane, and the central atom is then “capped” by 3 atoms on top, and 3 atoms below it.

23. Close Packing

24. # of Atoms/Unit Cell For atoms in a cubic unit cell: • Atoms in corners are ⅛ within the cell

25. # of Atoms/Unit Cell For atoms in a cubic unit cell: • Atoms on faces are ½ within the cell

26. # of Atoms/Unit Cell A face-centered cubic unit cell contains a total of 4 atoms: 1 from the corners, and 3 from the faces.

27. # of Atoms/Unit Cell For atoms in a cubic unit cell: • Atoms in corners are ⅛ within the cell • Atoms on faces are ½ within the cell • Atoms on edges are ¼ within the cell

28. Other Metallic Crystal Structures Body-centered cubic unit cells have an atom in the center of the cube as well as one in each corner. The packing efficiency is 68%, and the coordination number = 8.

29. Other Metallic Crystal Structures Simple cubic (or primitive cubic) unit cells are relatively rare. The atoms occupy the corners of a cube. The coordination number is 6, and the packing efficiency is only 52.4%.

30. Ex.iron system liquid 1538ºC -Fe BCC 1394ºC -Fe FCC 912ºC BCC -Fe Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism)  ex. titanium , -Ti ex. carbon (cubic)diamond, (hexagonal)graphite

31. Polymorphism Many metals exhibit different crystal structures with changes in pressure and temperature. It is temperature /pressure dependent phenomenon.

32. Example: Iron liquid above 1539 C. δ-iron (BCC) between 1394 and 1539 C. γ-iron (FCC) between 912 and 1394 C. α-iron (BCC) between -273 and 912 C.

33. Single crystal All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but they may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled.

34. Polycrystalline Materials • Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline • The Various stages in the solidification of a polycrystalline specimen are: • Initially, small crystals or nuclei form at various positions. • The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each. • The extremities of adjacent grains impinge on one another as the solidification process approaches completion • exists some atomic mismatch within the region where two grains meet; this area, called a grain boundary

36. Crystals as Building Blocks • Some engineering applications require single crystals: --turbine blades --diamond single crystals for abrasives • Properties of crystalline materials often related to crystal structure. --Ex: Quartz fractures more easily along some crystal planes than others.

37. Polycrystals Anisotropic • Most engineering materials are polycrystals. 1 mm Isotropic • Each "grain" is a single crystal. • If grains are randomly oriented, overall component properties are not directional. • Grain sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

38. Single vs Polycrystals E (diagonal) = 273 GPa E (edge) = 125 GPa • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: • Polycrystals 200 mm -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.

40. nA VCNA Mass of Atoms in Unit Cell Total Volume of Unit Cell  = Theoretical Density, r A knowledge of crystal structure of a metallic solid permits computation density : Density =  = where n = number of atoms/unit cell A =atomic weight g/mole VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.023 x 1023 atoms/mol

41. R a 2 52.00 atoms g  = unit cell mol atoms 3 a 6.023x1023 mol volume unit cell Theoretical Density, r • Ex: Cr (BCC) A =52.00 g/mol R = 0.125 nm n = 2 a = 4R/ 3 = 0.2887 nm theoretical = 7.18 g/cm3 ractual = 7.19 g/cm3

42. r r r metals ceramics polymers Platinum Gold, W Tantalum Silver, Mo Cu,Ni Steels Tin, Zinc Zirconia Titanium Al oxide Diamond Si nitride Aluminum Glass - soda Glass fibers Concrete PTFE Silicon GFRE* Carbon fibers Magnesium G raphite CFRE * Silicone A ramid fibers PVC AFRE * PET PC H DPE, PS PP, LDPE Wood Densities of Material Classes In general Graphite/ Metals/ Composites/ Ceramics/ Polymers > > Alloys fibers Semicond 30 Why? 2 0 *GFRE, CFRE, & AFRE are Glass, Metals have... • close-packing (metallic bonding) • often large atomic masses Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers 10 in an epoxy matrix). Ceramics have... • less dense packing • often lighter elements 5 3 4 (g/cm ) 3 r 2 Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O) 1 0.5 Composites have... • intermediate values 0.4 0.3

43. Energy and Packing Energy typical neighbor bond length typical neighbor r bond energy • Dense, ordered packing Energy typical neighbor bond length r typical neighbor bond energy • Non dense, random packing Dense, ordered packed structures tend to 1.have lower energies and more stable energy arrangements. 2. Atoms come closer together and bond more tightly

44. Attractive Energy: the energy released when the ions come close together Repulsive energy: the energy absorbed as the ions come close together Net Energy: the sum of energies associated with the attraction and repulsion of the ions It is minimum when the ions are at their equilibrium separation distance r0. At the minimum energy, the force between the ions are zero. The ions will remain at an equilibrium separation distance r0 ( more stable).

45. Zincblende/Diamond Lattices Diamond Lattice The Cubic Unit Cell Zincblende Lattice The Cubic Unit Cell Other views of the cubic unit cell

46. Diamond Lattice Diamond Lattice The Cubic Unit Cell

47. Zincblende (ZnS) Lattice Zincblende Lattice The Cubic Unit Cell.

48. Wurtzite Structure • We’ve also seen:Many semiconductors have the Wurtzite Structure Tetrahedral coordination:Each atom has 4 nearest-neighbors (nn).Basis set:2 atoms.Primitive lattice hexagonal close packed (hcp). 2 atoms per hcp lattice point A Unit Cell looks like

49. Materials and Packing Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Si Oxygen Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2

50. Crystal: is a solids in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. Long range order exist • Noncrystalline(amorphous): materials that don’t crystallize, this long range atomic order is absent • Crystal structure: it is the manner in which atoms, ions, or molecules are spatially arranged. Or • A regular 3 dimensional pattern of atoms or ions in space • There are an extremely large number of different crystal structures all having long range atomic order. • Crystal system: is described in terms of the unit cell geometry. • A crystal structure is described by both the geometry of, and atomic arrangements within the unit cell, whereas a • Crystal system is described only in terms of the unit cell geometry. For example, face-centered cubic and body-centered cubic are crystal structures that belong to the cubic crystal system.