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. 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 have lower energies.

3. Materials and Packing Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Adapted from Fig. 3.23(a), Callister & Rethwisch 8e. Si Oxygen Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2 Adapted from Fig. 3.23(b), Callister & Rethwisch 8e.

4.  Metallic Crystal Structures • How can we stack metal atoms to minimize empty space? 2-dimensions vs. Now stack these 2-D layers to make 3-D structures

5. Metallic Crystal Structures • Tend to be densely packed. • Reasons for dense packing: - Typically, only one element is present, so all atomic radii are the same. - Metallic bonding is not directional. - Nearest neighbor distances tend to be small in order to lower bond energy. - Electron cloud shields cores from each other • Have the simplest crystal structures. We will examine three such structures...

6. Simple Cubic Structure (SC) • Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors) Click once on image to start animation (Courtesy P.M. Anderson)

7. Atomic Packing Factor (APF) atoms 4 a 3 unit cell p (0.5a) 3 R=0.5a volume close-packed directions unit cell contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.24, Callister & Rethwisch 8e. Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres • APF for a simple cubic structure = 0.52 volume atom 1 APF = 3 a

8. Body Centered Cubic Structure (BCC) • Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Cr, W, Fe (), Tantalum, Molybdenum • Coordination # = 8 Adapted from Fig. 3.2, Callister & Rethwisch 8e. Click once on image to start animation (Courtesy P.M. Anderson) 2 atoms/unit cell: 1 center + 8 corners x 1/8

9. Atomic Packing Factor: BCC a 3 a 2 Close-packed directions: R 3 a length = 4R = a atoms volume 4 3 p ( 3 a/4 ) 2 unit cell atom 3 APF = volume 3 a unit cell • APF for a body-centered cubic structure = 0.68 a Adapted from Fig. 3.2(a), Callister & Rethwisch 8e.

10. Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Au, Pb, Ni, Pt, Ag • Coordination # = 12 Adapted from Fig. 3.1, Callister & Rethwisch 8e. Click once on image to start animation 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8 (Courtesy P.M. Anderson)

11. Atomic Packing Factor: FCC 2 a Unit cell contains: 6 x1/2 + 8 x1/8 = 4 atoms/unit cell a atoms volume 4 3 p ( 2 a/4 ) 4 unit cell atom 3 APF = volume 3 a unit cell • APF for a face-centered cubic structure = 0.74 maximum achievable APF! Close-packed directions: 2 a length = 4R = Adapted from Fig. 3.1(a), Callister & Rethwisch 8e.

12. B B B B C C C A A A B B B B B B B A sites C C C C C C B B sites sites B B B B C sites FCC Stacking Sequence • ABCABC... Stacking Sequence • 2D Projection A • FCC Unit Cell B C

13. HCP Stacking Sequence

14. Hexagonal Close-Packed Structure (HCP) A sites Top layer c Middle layer B sites A sites Bottom layer a • ABAB... Stacking Sequence • 3D Projection • 2D Projection Adapted from Fig. 3.3(a), Callister & Rethwisch 8e. 6 atoms/unit cell • Coordination # = 12 ex: Cd, Mg, Ti, Zn • APF = 0.74 • c/a = 1.633

15. nA VCNA Mass of Atoms in Unit Cell Total Volume of Unit Cell  = Theoretical Density, r Density =  = where n = number of atoms/unit cell A =atomic weight VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.022 x 1023 atoms/mol

16. R a 2 52.00 atoms g  = unit cell mol atoms a3 6.022x1023 mol volume unit cell Theoretical Density, r • Ex: Cr (BCC) A =52.00 g/mol R = 0.125 nm n = 2 atoms/unit cell a = 4R/ 3 = 0.2887 nm Adapted from Fig. 3.2(a), Callister & Rethwisch 8e. theoretical = 7.18 g/cm3 ractual = 7.19 g/cm3

17. 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? B ased on data in Table B1, Callister 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 Data from Table B.1, Callister & Rethwisch, 8e.

18. SINGLE CRYSTALS f16_03_pg64 Garnet single crystal from China Ruby crystals from Tanzania Pyrite (FeS2) (Fool’s Gold) crystals from Spain

19. Crystals as Building Blocks • Some engineering applications require single crystals: -- turbine blades -- diamond single crystals for abrasives Fig. 8.33(c), Callister & Rethwisch 8e. (Fig. 8.33(c) courtesy of Pratt and Whitney). (Courtesy Martin Deakins, GE Superabrasives, Worthington, OH. Used with permission.) • Properties of crystalline materials often related to crystal structure. -- Ex: Quartz fractures more easily along some crystal planes than others. (Courtesy P.M. Anderson)

20. Polycrystals • Most crystalline solids are composed of many small crystals (also called grains). • Initially, small crystals (nuclei) form at various positions. • These have random orientations. • The small grains grow and begin to impinge on one another forming grain boundaries.

21. POLYCRYSATLLINE MATERIALS

22. Polycrystals Anisotropic • Most engineering materials are polycrystals. Adapted from Fig. K, color inset pages of Callister 5e. (Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany) 1 mm Isotropic • Nb-Hf-W plate with an electron beam weld. • Each "grain" is a single crystal. • If grains are randomly oriented, overall component properties are not directional. • Grain sizes typically range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

23. Single vs Polycrystals E (diagonal) = 273 GPa Data from Table 3.3, Callister 7e. (Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.) E (edge) = 125 GPa 200 mm • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: • Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic. Adapted from Fig. 4.14(b), Callister 7e. (Fig. 4.14(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)

25. Fig. 3.4, Callister & Rethwisch 8e. Crystal Systems Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems 14 crystal lattices a, b, and c are the lattice constants

27. (c) 2003 Brooks/Cole Publishing / Thomson Learning™ The fourteen (14) types of Bravais lattices grouped in seven (7) systems.

28. z 111 c y 000 b a x Point Coordinates Point coordinates for unit cell center are a/2, b/2, c/2 ½½½ Point coordinates for unit cell corner are 111 Translation: integer multiple of lattice constants  identical position in another unit cell z 2c y b b

29. [111] where overbar represents a negative index => Crystallographic Directions Algorithm z 1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of unit cell dimensions a, b, and c3. Adjust to smallest integer values4. Enclose in square brackets, no commas [uvw] y x ex:1, 0, ½ => 2, 0, 1 => [201] -1, 1, 1 families of directions <uvw>

30. [110] # atoms 2 a = = 1 - LD 3.5 nm 2 a length Linear Density Number of atoms Unit length of direction vector • Linear Density of Atoms  LD =  ex: linear density of Al in [110] direction a = 0.405 nm Adapted from Fig. 3.1(a), Callister & Rethwisch 8e.

31. Crystallographic Planes Adapted from Fig. 3.10, Callister & Rethwisch 8e.

32. Crystallographic Planes • Miller Indices: Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices. • Algorithm  1.  Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl)

33. example example abc abc z 1 1  1. Intercepts c 1/1 1/1 1/ 2. Reciprocals 1 1 0 3. Reduction 1 1 0 y b a z x 1/2   1. Intercepts c 1/½ 1/ 1/ 2. Reciprocals 2 0 0 3. Reduction 2 0 0 y b a x Crystallographic Planes 4. Miller Indices (110) 4. Miller Indices (200)

34. z c 1/2 1 3/4 1. Intercepts 1/½ 1/1 1/¾ 2. Reciprocals 2 1 4/3 y b a 3. Reduction 6 3 4 x Family of Planes {hkl} Ex: {100} = (100), (010), (001), (100), (001) (010), Crystallographic Planes example a b c 4. Miller Indices (634)

35. 2D repeat unit (100) 4 3 = a R 3 Radius of iron R = 0.1241 nm Adapted from Fig. 3.2(c), Callister 7e. atoms 2 2D repeat unit a atoms 1 atoms 1 12.1 = 1.2 x 1019 2 = = nm2 m2 4 3 R 3 area 2D repeat unit Planar Density of (100) Iron Solution:  At T < 912C iron has the BCC structure. Planar Density =

36. FCC Unit Cell with (110) plane

37. BCC Unit Cell with (110) plane

38. X-Ray Diffraction • Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation. • Can’t resolve spacings  • Spacing is the distance between parallel planes of atoms.  

40. X-ray nl intensity q d= c 2sin (from detector) q q c X-Rays to Determine Crystal Structure • Incoming X-rays diffract from crystal planes. Measurement of critical angle, qc, allows computation of planar spacing, d.

41. X-RAY DIFFRACTOMETER

42. z z z c c c y y y a a a b b b x x x X-Ray Diffraction Pattern (110) (211) Intensity (relative) (200) Diffraction angle 2q Diffraction pattern for polycrystalline a-iron (BCC) Adapted from Fig. 3.22, Callister 8e.

43. SAMPLE PROBLEM

44. Allotropy or Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism). • Allotropy (Gr. allos, other, and tropos, manner) or allotropism is a behavior exhibited by certain chemical elements that can exist in two or more different forms, known as allotropes of that element. • In each allotrope, the element's atoms are bonded together in a different manner. • Note that allotropy refers only to different forms of an element within the same phase or state of matter (i.e. different solid, liquid or gas forms) - the changes of state between solid, liquid and gas in themselves are not considered allotropy.

45. 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)  titanium , -Ti carbon diamond, graphite

46. Polymorphic Forms of Carbon Diamond and graphite are two allotropes of carbon: pure forms of the same element that differ in structure. Diamond • An extremely hard, transparent crystal with • tetrahedral bonding of carbon • very high thermal conductivity • very low electric conductivity. • The large single crystals are typically used as gem stones • The small crystals are used to grind/cut other materials • diamond thin films • hard surface coatings – used for cutting tools, medical devices.

47. Polymorphic Forms of Carbon Graphite • a soft, black, flaky solid, with a layered structure – parallel hexagonal arrays of carbon atoms • weak van der Waal’s forces between layers • planes slide easily over one another

48. Polymorphic Forms of Carbon Fullerenes and Nanotubes • Fullerenes – spherical cluster of 60 carbon atoms, C60 • Like a soccer ball • Carbon nanotubes – sheet of graphite rolled into a tube • Ends capped with fullerene hemispheres

49. Tin, Allotropic Transformation Volume increase of 27% Density decrease from 7.3 to 5.77 g/cm3 Tin Disease: Another common metal that experiences an allotropic change is tin. White tin (BCT) transforms into gray tin (diamond cubic) at 13.2 ˚C. The transformation rate is slow, though, the lower the temp, the faster the rate. Due to the volume expansion, the metal disintegrates into the coarse gray powder. In 1850 Russia, a particularly cold winter crumbled the tin buttons on some soldier uniforms. There were also problems with tin church organ pipes.

50. SUMMARY • Atoms may assemble into crystalline or amorphous structures. • Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures. • We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). • Crystallographic points, directions and planes are specified in terms of indexing schemes.