References:

1. CHAPTER 3 STRUCTURES OF METALS AND CERAMICS References: • Gregory S. Rohrer, Structure and Bonding in Crystalline Materials (Cambridge University Press, 2004) • William D. Callister, Jr., Fundamentals of Materials Science and Engineering 5th ed. (John Wiley and Sons Inc., 2001) ES 67 : ELEMENTS OF MATERIAL SCIENCE AND ENGINEERING Maria Cristina P. Vegafria

2. What is crystallography? Originated as the study of macroscopic crystal forms Modern crystallography has been redefined by X-ray Diffraction. Its primary concern is with the study of atomic arrangements in crystalline materials.

3. Processing Why study crystallography? Structure Properties Performance Materials Science and Engineering

4. Fundamental Concepts CRYSTAL “A region of matter within which the atoms are arranged in a three-dimensional translationally periodic pattern.” -Buerger (1956)

5. Fundamental Concepts Crystal Structure- the manner in which atoms, ions, or molecules are spatially arranged.

6. Fundamental Concepts Crystalline Material- one in which the atoms are situated in a repeating or periodic array over large atomic distances. • All metals, many ceramic materials, and certain polymers form crystalline structures under normal solidification conditions. Potassium dihydrogen phosphate

7. Fundamental Concepts Materials that do not crystallize are callednoncrsytallineoramorphousmaterials.

8. Unit Cells Unit Cell • the smallest structural unit or building block that can describe the crystal structure. Repetition of the unit cell generates the entire crystal. • smallest repetitive volume which contains the complete lattice pattern of a crystal. Unit cells for most crystal structures are parallelepipeds or prisms having three sets of parallel faces.

9. Fundamental Concepts Crystal Lattice • a three-dimensional array of points coinciding with atom positions (or sphere centers) • the periodic and systematic arrangement of atoms that are found in crystals with the exception of amorphous solids and gases • can be considered as the points of intersection between straight lines in a three-dimensional network

10. Crystalline and Noncrystalline Materials Single Crystals For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extends throughout the entirety of the specimen without interruption, the result is a single crystal. All unit cells interlock in the same way and have the same orientation. If the extremities of a single crystal are permitted to grow without any external constraint, the crystal will assume a regular geometric shape having flat faces, as with some of the gem stones; the shape is indicative of the crystal structure.

11. Crystalline and Noncrystalline Materials Polycrystalline Materials Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline.

12. Atomic Hard Sphere Model • Atoms (or ions) are thought of as being solid spheres having well-defined diameters • Spheres representing nearest- neighbour atoms touch one another.

13. Metallic Crystal Structures • Atomic bonding is metallic, which is nondirectional in nature • No restrictions to the number and position of nearest-neighbor atoms • Large number of nearest neighbors and dense atomic packings

14. Two other important characteristics of a crystal structure are the coordination number and the atomic packing factor (APF). Coordination Number – the number of nearest-neighbor or touching atoms APF- the fraction of solid sphere volume in a unit cell

15. 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)

16. 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 atomic radius

17. Metallic Crystal Structures • Face-Centered Cubic (FCC) • Body-Centered Cubic (BCC) • Hexagonal Close-Packed (HCP)

18. Face-Centered Cubic Crystal Structure Atoms are located at each of the corners and at the centers of all the cube faces. • Atoms touch each other along face diagonals. • Coordination # = 12 4 atoms/unit cell: (6 face x ½) + (8 corners x 1/8) Examples : copper, aluminum, silver and gold.

19. 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! Adapted from Fig. 3.1(a), Callister 7e.

20. Body-Centered Cubic (BCC) Crystal Structure Atoms are located at all eight corners and a single atom at the cube center. • Atoms touch each other along cube diagonals within a unit cell. 2 atoms/unit cell: (1 center) + (8 corners x 1/8) • Coordination # = 8 Examples: chromium, iron (), Tantalum, molybdenum,, tungsten

21. 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

22. A sites c B sites A sites a Hexagonal Close-Packed (HCP) Crystal Structure • Coordination # = 12 6 atoms/unit cell • c/a = 1.633 (ideal) • APF = 0.74 Examples: cadmium, magnesium, titanium, and zinc.

23. Close-Packed Crystal Structures (Metals) Let the centers of all the atoms in one close-packed plane be labeled A. Associated with this plane are two sets of equivalent triangular depressions formed by three adjacent atoms, into which the next close-packed plane of atoms may rest. Those having the triangle vertex pointing up are arbitrarily designated as B positions, while the remaining depressions are those with the down vertices, which are marked C. A second close-packed plane may be positioned with the centers of its atoms over either B or C sites; at this point both are equivalent. If B is chosen, the stacking sequence is termed AB.

24. Hexagonal Close – Packed The distinction between FCC and HCP lies in where the third close-packed layer is positioned. For HCP, the centers of this layer are aligned directly above the original A positions. This stacking sequence, ABABAB . . . , is repeated over and over. Of course, the ACACAC . . . arrangement would be equivalent.

25. Face –Centered Cubic For the face-centered crystal structure, the centers of the third plane are situated over the C sites of the first plane. This yields an ABCABCABC . . . stacking sequence; that is, the atomic alignment repeats every third plane. These planes are of the (111) type.

26. Metallic Crystal Structures

28. Lattice Structures Part 1.flv

29. Crystal Systems An x, y, z coordinate system is established with its origin at one of the unit cell corners; each of the x, y, z axes coincides with one of the three parallelepiped edges that extends from the corner. The unit cell geometry is completely defined in terms of six parameters: the three edge lengths a, b, and c, and the three interaxial angles α, β, and γ.

30. The Seven Crystal Systems • Cubic • Hexagonal • Tetragonal • Rhombohedral • Orthorhombic • Monoclinic • Triclinic

33. The 14 Bravais Lattices

34. The+fourteen+Bravais+Lattices.flv

35. Crystallographic Directions and Planes • Labeling conventions have been established in which three integers or indices are used to designate directions and planes. • The basis for determining index values is the unit cell, with a coordinate system consisting of three (x, y, and z) axes situated at one of the corners and coinciding with the unit cell edges.

36. 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 z

37. Crystallographic Directions A crystallographic direction is defined as a line between two points, or a vector. The following steps are utilized in the determination of the three directional indices: • A vector of convenient length is positioned such that it passes through the origin of the coordinate system. Any vector may be translated throughout the crystal lattice without alteration, if parallelism is maintained. • The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a, b, and c.

38. Crystallographic Directions • These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values. • The three indices, not separated by commas, are enclosed in square brackets, thus: [uvw]. The u, v, and w integers correspond to the reduced projections along the x, y, and z axes, respectively. • For each of the three axes, there will exist both positive and negative coordinates. • Thus negative indices are also possible, which are represented by a bar over the appropriate index.

39. 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]

40. Crystallographic Directions For some crystal structures, several nonparallel directions with different indices are actually equivalent. This means that the spacing of atoms along each direction is the same. For example, in cubic crystals, all the directions represented by the following indices are equivalent: [100], [100], [010], [010], [001], and [001].

41. Crystallographic Directions As a convenience, equivalent directions are grouped together into a family, which are enclosed in angle brackets, thus: <100>. Furthermore, directions in cubic crystals having the same indices without regard to order or sign, for example, [123] and [213], are equivalent. This is, in general, not true for other crystal systems.

42. Linear Atomic Density Directional equivalency is related to the atomic linear density in the sense that equivalent directions have identical linear densities. The fraction of the line length intersected by these atoms is equal to the linear density. The direction vector is positioned so as to pass through atom centers,

43. [110] # atoms 2 a - = = 1 LD 3.5 nm 2 a length 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

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

45. Hexagonal Crystals A problem arises for crystals having hexagonal symmetry in that some crystallographic equivalent directions will not have the same set of indices. This is circumvented by utilizing a four-axis, or Miller–Bravais, coordinate system. Coordinate axis system for a hexagonal unit cell (Miller–BravaisScheme).

46. Hexagonal Crystals The three a1 , a2 , and a3 axes are all contained within a single plane (called the basal plane), and at 120 angles to one another. The z axis is perpendicular to this basal plane. Coordinate axis system for a hexagonal unit cell (Miller–BravaisScheme).

47. Hexagonal Crystals Directional indices, which are obtained as described above, will be denoted by four indices, as [uvtw];by convention, the first three indices pertain to projections along the respective a1 , a2 , and a3 axes in the basal plane.

48. Hexagonal Crystals Conversion from the three-index system to the four-index system, [u’v’w’] [uvtw] where primed indices are associated with the three-index scheme and unprimed, with the new Miller–Bravais four-index system; nis a factor that may be required to reduce u, v, t, and w to the smallest integers. For example, using this conversion, the [010] direction becomes

49. Crystallographic Planes • The orientations of planes for a crystal structure are represented in a similar manner. A ‘‘family’’ of planes contains all those planes that are crystallographically equivalent—that is, having the same atomic packing; and a family is designated by indices that are enclosed in braces—e.g., {100}.

50. Crystallographic Planes Miller Indices A Miller Index is a series of coprime integers that are inversely proportional to the intercepts of the crystal face or crystallographic planes with the edges of the unit cell.  It describes the orientation of a plane in the 3-D lattice with respect to the axes. The general form of the Miller index is (h, k, l) where h, k, and l are integers related to the unit cell along the a, b, c crystal axes.