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Crystallography and Structure

Crystallography and Structure. ME 2105 R. R. Lindeke. Overview:. Crystal Structure – matter assumes a periodic shape Non-Crystalline or Amorphous “structures” no long range periodic shapes FCC, BCC and HCP – common for metals Xtal Systems – not structures but potentials

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Crystallography and Structure

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  1. Crystallography and Structure ME 2105 R. R. Lindeke

  2. Overview: • Crystal Structure – matter assumes a periodic shape • Non-Crystalline or Amorphous “structures” no long range periodic shapes • FCC, BCC and HCP – common for metals • Xtal Systems – not structures but potentials • Point, Direction and Planer ID’ing in Xtals • X-Ray Diffraction and Xtal Structure

  3. 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.

  4. CRYSTAL STRUCTURES Crystal Structure • Means: PERIODIC ARRANGEMENT OF ATOMS/IONS • OVER LARGE ATOMIC DISTANCES • Leads to structure displayingLONG-RANGE ORDER that is Measurable and Quantifiable All metals, many ceramics, some polymers exhibit this “High Bond Energy” – More Closely Packed Structure

  5. Materials Lacking Long range order Amorphous Materials These less densely packed lower bond energy “structures” can be found in Metal are observed in Ceramic GLASS and many “plastics”

  6. Fig. 3.4, Callister 7e. Crystal Systems – Some Definitional information Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems of varying symmetry are known These systems are built by changing the lattice parameters:a, b, and c are the edge lengths , , and  are interaxial angles

  7. Crystal Systems Crystal structures are divided into groups according to unit cell geometry (symmetry).

  8. 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 (those of engineering importance) called: FCC, BCC and HCP – with a nod to Simple Cubic

  9. Crystal Structure of Metals – of engineering interest

  10. Simple Cubic Structure (SC) • Rare due to low packing density (only Po – Polonium -- has this structure) • Close-packed directions are cube edges. • Coordination No. = 6 (# nearest neighbors) for each atom as seen (Courtesy P.M. Anderson)

  11. Atomic Packing Factor (APF) volume atoms atom 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. 3.23, Callister 7e. Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres • APF for a simple cubic structure = 0.52 1 APF = 3 a Here: a = Rat*2 Where Rat is the ‘handbook’ atomic radius

  12. 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 7e. 2 atoms/unit cell: (1 center) + (8 corners x 1/8) (Courtesy P.M. Anderson)

  13. 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 a Adapted from Fig. 3.2(a), Callister 7e. • APF for a body-centered cubic structure = 0.68

  14. 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 7e. 4 atoms/unit cell: (6 face x ½) + (8 corners x 1/8) (Courtesy P.M. Anderson)

  15. Atomic Packing Factor: FCC Close-packed directions: 2 a length = 4R = 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 The maximum achievable APF! (a = 22*R) Adapted from Fig. 3.1(a), Callister 7e.

  16. Hexagonal Close-Packed Structure (HCP) A sites Top layer c Middle layer B sites A sites Bottom layer a ex: Cd, Mg, Ti, Zn • ABAB... Stacking Sequence • 3D Projection • 2D Projection Adapted from Fig. 3.3(a), Callister 7e. 6 atoms/unit cell • Coordination # = 12 • APF = 0.74 • c/a = 1.633 (ideal)

  17. We find that both FCC & HCP are highest density packing schemes (APF = .74) – this illustration shows their differences as the closest packed planes are “built-up”

  18. 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 = 6.023 x 1023 atoms/mol

  19. R a 2 52.00 theoretical atoms = 7.18 g/cm3 g  = unit cell ractual = 7.19 g/cm3 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

  20. Adding Ionic Complexities Looking at the Ceramic Unit Cells

  21. Cesium chloride (CsCl) unit cell showing (a) ion positions and the two ions per lattice point and (b) full-size ions. Note that the Cs+−Cl− pair associated with a given lattice point is not a molecule because the ionic bonding is nondirectional and because a given Cs+is equally bonded to eight adjacent Cl−, and vice versa. [Part (b) courtesy of Accelrys, Inc.]

  22. Sodium chloride (NaCl) structure showing (a) ion positions in a unit cell, (b) full-size ions, and (c) many adjacent unit cells. [Parts (b) and (c) courtesy of Accelrys, Inc.]

  23. Fluorite (CaF2) unit cell showing (a) ion positions and (b) full-size ions. [Part (b) courtesy of Accelrys, Inc.]

  24. iron system: liquid 1538ºC -Fe BCC 1394ºC -Fe FCC 912ºC BCC -Fe Polymorphism: Also in Metals • Two or more distinct crystal structures for the same material (allotropy/polymorphism)  titanium  (HCP), (BCC)-Ti carbon: diamond, graphite

  25. The corundum (Al2O3) unit cell is shown superimposed on the repeated stacking of layers of close-packed O2−ions. The Al3+ions fill two-thirds of the small (octahedral) interstices between adjacent layers.

  26. Exploded view of the kaolinite unit cell, 2(OH)4Al2Si2O5. (From F. H. Norton, Elements of Ceramics, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, MA, 1974.)

  27. Transmission electron micrograph of the structure of clay platelets. This microscopic-scale structure is a manifestation of the layered crystal structure shown in the previous slide. (Courtesy of I. A. Aksay.)

  28. (a) An exploded view of the graphite (C) unit cell. (From F. H. Norton, Elements of Ceramics, 2nd ed., Addison-Wesley Publishing Co., Inc., Reading, MA, 1974.) (b) A schematic of the nature of graphite’s layered structure. (From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics, 2nd ed., John Wiley & Sons, Inc., NY, 1976.)

  29. (a) C60molecule, or buckyball. (b) Cylindrical array of hexagonal rings of carbon atoms, or buckytube. (Courtesy of Accelrys, Inc.)

  30. Arrangement of polymeric chains in the unit cell of polyethylene. The dark spheres are carbon atoms, and the light spheres are hydrogen atoms. The unit-cell dimensions are 0.255 nm × 0.494 nm × 0.741 nm. (Courtesy of Accelrys, Inc.)

  31. Weaving-like pattern of folded polymeric chains that occurs in thin crystal platelets of polyethylene. (From D. J. Williams, Polymer Science and Engineering, Prentice Hall, Inc., Englewood Cliffs, NJ, 1971.)

  32. Diamond cubic unit cell showing (a) atom positions. There are two atoms per lattice point (note the outlined example). Each atom is tetrahedrally coordinated. (b) The actual packing of full-size atoms associated with the unit cell. [Part (b) courtesy of Accelrys, Inc.]

  33. Zinc blende (ZnS) unit cell showing (a) ion positions. There are two ions per lattice point (note the outlined example). Compare this structure with the diamond cubic structure (Figure 3.20a). (b) The actual packing of full-size ions associated with the unit cell. [Part (b) courtesy of Accelrys, Inc.]

  34. 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 Data from Table B1, Callister 7e.

  35. Crystals as Building Blocks • Some engineering applications require single crystals: --diamond single crystals for abrasives --turbine blades Fig. 8.33(c), Callister 7e. (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)

  36. 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 typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

  37. Single vs Polycrystals E (diagonal) = 273 GPa E (edge) = 125 GPa • Single Crystals 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.) -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. 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].)

  38. Effects of Anisotropy:

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

  40. [111] where ‘overbar’ represents a negative index => Crystallographic Directions Algorithm z 1. Vector is repositioned (if necessary) to pass through the Unit Cell origin.2. Read off line projections (to principal axes of U.C.) 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>

  41. xy z Projections: Projections in terms of a,b and c: Reduction: Enclosure [brackets] What is this Direction ????? 0c a/2 b 0 1 1/2 2 0 1 [120]

  42. [110] # atoms 2 a - = = 1 LD 3.5 nm 2 a length Linear Density – considers equivalance and is important in Slip 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 # atoms CENTERED on the direction of interest! Length is of the direction of interest within the Unit Cell

  43. Determining Angles Between Crystallographic Direction: Where ui’s , vi’s & wi’s are the “Miller Indices” of the directions in question – also (for information) If a direction has the same Miller Indicies as a plane, it is NORMAL to that plane

  44. z a2 - a2 a3 -a3 a 2 a1 2 a3 ex: ½, ½, -1, 0 => [1120] a 1 2 dashed red lines indicate projections onto a1 and a2 axes a1 HCP Crystallographic Directions Algorithm 1. Vector repositioned (if necessary) to pass through origin.2. Read off projections in terms of unit cell dimensions a1, a2, a3, or c3. Adjust to smallest integer values4. Enclose in square brackets, no commas [uvtw] Adapted from Fig. 3.8(a), Callister 7e.

  45. z ® u [ ' v ' w ' ] [ uvtw ] 1 = - u ( 2 u ' v ' ) 3 a3 a2 1 = - v ( 2 v ' u ' ) a1 3 - = - + t ( u v ) = w w ' Fig. 3.8(a), Callister 7e. HCP Crystallographic Directions • Hexagonal Crystals • 4 parameter Miller-Bravais lattice coordinates are related to the direction indices (i.e., u'v'w') in the ‘3 space’ Bravais lattice as follows.

  46. z a3 a2 a1 - Fig. 3.8(a), Callister 7e. Computing HCP Miller- Bravais Directional Indices (an alternative way): We confine ourselves to the bravais parallelopiped in the hexagon: a1-a2-Z and determine: (u’,v’w’) Here: [1 1 0] - so now apply the models to create M-B Indices

  47. Defining 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 (cubic lattices is direct) 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) families {hkl}

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