![Close packed crystals
A plane
B plane
C plane
A plane
…ABCABCABC… packing
[Face Centered Cubic (FCC)]
…ABABAB… packing
[Hexagonal Close Packing (HCP)]](https://image.slidesharecdn.com/crystalsystemlec3pdf-201029071814/85/Crystal-and-Crystal-Systems-PowerPoint-Presentation-26-320.jpg)
![CRYSTALLOGRAPHIC DIRECTIONS
• Defined as line between two points: a vector
• Steps for finding the 3 indices denoting a direction
– Determine the point positions of a beginning point (X1 Y1 Z1) and a
ending point (X2 Y2 Z2) for direction, in terms of unit cell edges
– Calculate difference between ending and starting point
– Multiply the differences by a common constant to convert them to the
smallest possible integers u, v, w
– The three indices are not separated by commas and are enclosed in
square brackets: [uvw]
– If any of the indices is negative, a bar is placed in top of that index](https://image.slidesharecdn.com/crystalsystemlec3pdf-201029071814/85/Crystal-and-Crystal-Systems-PowerPoint-Presentation-33-320.jpg)
![EXAMPLES: DIRECTIONS
• Draw a [1,-1,0] direction within a cubic unit cell
• Determine the indices for this direction
– Answer: [120]](https://image.slidesharecdn.com/crystalsystemlec3pdf-201029071814/85/Crystal-and-Crystal-Systems-PowerPoint-Presentation-35-320.jpg)
![Family of planes and directions
• Miller indices of the specific plane (hkl).
• Family of planes {hkl}
• Crystallographic direction are indicated by
[uvw]
• Family of Crystallographic directions are
indicated by <uvw>](https://image.slidesharecdn.com/crystalsystemlec3pdf-201029071814/85/Crystal-and-Crystal-Systems-PowerPoint-Presentation-42-320.jpg)
Crystal and Crystal Systems PowerPoint Presentation
- 2. What is a crystal • Something is crystalline if the atoms or ions that compose it are arranged in a regular way i.e • A crystal has internal order due to the periodic arrangement of atoms in three dimensions. • A crystal is built up by arranging atoms and groups of atoms in regular patterns, • The basic arrangement of atoms that describes the crystal structure is identified and is called unit cell.
- 3. Infinite three dimensional array of points. Every point has identical surrounding
- 4. Space Lattice • Infinite three dimensional array of points in which every point has identical surrounding . • These points with identical surroundings are called lattice points • Lattice points can be arranged in 14 different arrays, called bravais lattice.
- 6. SOME DEFINITIONS … • Lattice: 3D array of regularly spaced points • Crystalline material: atoms situated in a repeating 3D periodic array over large atomic distances • Amorphous material: material with no such order • Hard sphere representation: atoms denoted by hard, touching spheres • Unit cell: basic building block unit that repeats in space to create the crystal structure
- 7. lattice constant, or lattice parameter • Refers to the constant distance between unit cells in a crystal lattice. • Lattices in three dimension have three lattice constants, referred to as a, b, and c. • However, in the special case of cubic crystal structures, all of the constants are equal and we only refer to a. • Similarly, in hexagonal crystal structures, the a and b constants are equal, and we only refer to the a and c constants. • A group of lattice constants could be referred to as lattice parameters. However, the full set of lattice parameters consist of the three lattice constants and the three angles between them.
- 8. CRYSTAL SYSTEMS • Based on shape of unit cell ignoring actual atomic locations • Unit cell = 3-dimensional unit that repeats in space • Unit cell geometry completely specified by a, b, c & a, b, g (lattice parameters or lattice constants) • Seven possible combinations of a, b, c & a, b, g, resulting in seven crystal systems
- 9. Metal • A metal is a solid material that is typically hard, opaque, shiny, • Features good electrical and thermal conductivity. • Metals are generally malleable and ductile • That is, they can be hammered or pressed and drawn out into a thin wire permanently out of shape without breaking or cracking • Able to be fused or melted
- 10. Metal • Many questions about metal can be answered by knowing their atomic structure • The arrangement of the atoms within the metals. • Arrangement of the atoms in a metal, which are held together by metallic bond to form a close-packed regular giant structure. • The free electrons between the atoms enable metals to conduct heat and electricity • The high melting points of metals suggest that there are strong chemical bonds between the atoms.
- 11. Crystalline Material • Metals and many non-metallic solids are crystalline i.e • The constituent atoms are arranged in a pattern that repeat itself periodically in three dimensions. • The actual arrangement of the atoms is described by the crystal structure.
- 12. • The crystal structures of most pure metals are simple. • Three most common structures are: • Body-centered cubic, • Face-centered cubic, • Closed-packed hexagonal structures • In contrast, the structures of alloys and non- metallic compounds are often complex.
- 13. • Cubic unit cell is 3D repeat unit • Rare (only Polonium has this structure) • Close-packed directions (directions along which atoms touch each other) are cube edges. • Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson) SIMPLE CUBIC STRUCTURE (SC)
- 14. 2of 4 basic atomic arrangements 2. Body-centered cubic (bcc) 1. A portion of the structure of a body-centered cubic metal (b.c.c.)
- 15. 3 of 4 basic atomic arrangements 3. Face-centered cubic (fcc) also known as Cubic close packing
- 16. 4 of 4 basic atomic arrangements 4. Hexagonal close-packed (hcp)
- 17. Review of the three basic Atomic Structures B.C.C. F.C.C H.C.P
- 18. • Cubic unit cell is 3D repeat unit • Rare (only Po has this structure) • Close-packed directions (directions along which atoms touch each other) are cube edges. • Coordination # = 6 (# nearest neighbors) (Courtesy P.M. Anderson) SIMPLE CUBIC STRUCTURE (SC)
- 19. ATOMIC PACKING FACTOR • Fill a box with hard spheres – Packing factor = total volume of spheres in box / volume of box – Question: what is the maximum packing factor you can expect? • In crystalline materials: – Atomic packing factor = total volume of atoms in unit cell / volume of unit cell
- 20. • APF for a simple cubic structure = 0.52 A P F for Simple Cubic Structure contains 8 x 1/8 = 1 atom/unit cell Adapted from Fig. 3.19, Callister 6e. Lattice constant close-packed directions a R=0.5a
- 21. • ABCABC... Stacking Sequence • FCC Unit Cell FCC STACKING SEQUENCE A sites B sites C sites B B B BB B B C C C A A • 2D Projection
- 22. • Coordination # = 12 Adapted from Fig. 3.1(a), Callister 6e.(Courtesy P.M. Anderson) • Close packed directions are face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. FACE CENTERED CUBIC STRUCTURE (FCC)
- 23. Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell a • APF for a body-centered cubic structure = p/(32) = 0.74 (best possible packing of identical spheres) Adapted from Fig. 3.1(a), Callister 6e. ATOMIC PACKING FACTOR: FCC
- 24. • Coordination # = 8 Adapted from Fig. 3.2, Callister 6e.(Courtesy P.M. Anderson) • Close packed directions are cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. BODY CENTERED CUBIC STRUCTURE (BCC)
- 25. a R • APF for a body-centered cubic structure = p3/8 = 0.68 Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell Adapted from Fig. 3.2, Callister 6e. ATOMIC PACKING FACTOR: BCC
- 26. Close packed crystals A plane B plane C plane A plane …ABCABCABC… packing [Face Centered Cubic (FCC)] …ABABAB… packing [Hexagonal Close Packing (HCP)]
- 27. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP) Ideally, c/a = 1.633 for close packing However, in most metals, c/a ratio deviates from this value
- 28. • Coordination # = 12 • ABAB... Stacking Sequence • APF = 0.74, for ideal c/a ratio of 1.633 • 3D Projection • 2D Projection A sites B sites A sites Adapted from Fig. 3.3, Callister 6e. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)
- 29. COMPARISON OF CRYSTAL STRUCTURES Crystal structure packing factor • Simple Cubic (SC) 0.52 • Body Centered Cubic (BCC) 0.68 • Face Centered Cubic (FCC) 0.74 • Hexagonal Close Pack (HCP) 0.74
- 30. CRYSTALLOGRAPHIC POINTS, DIRECTIONS & PLANES • In crystalline materials, often necessary to specify points, directions and planes within unit cell and in crystal lattice • Three numbers (or indices) used to designate points, directions (lines) or planes, based on basic geometric notions • The three indices are determined by placing the origin at one of the corners of the unit cell, and the coordinate axes along the unit cell edges
- 31. POINT COORDINATES • Any point within a unit cell specified as fractional multiples of the unit cell edge lengths • Position P specified as q r s; convention: coordinates not separated by commas or punctuation marks
- 32. EXAMPLE: POINT COORDINATES • Locate the point (1/4 1 ½) • Specify point coordinates for all atom positions for a BCC unit cell – Answer: 0 0 0, 1 0 0, 1 1 0, 0 1 0, ½ ½ ½, 0 0 1, 1 0 1, 1 1 1, 0 1 1
- 33. CRYSTALLOGRAPHIC DIRECTIONS • Defined as line between two points: a vector • Steps for finding the 3 indices denoting a direction – Determine the point positions of a beginning point (X1 Y1 Z1) and a ending point (X2 Y2 Z2) for direction, in terms of unit cell edges – Calculate difference between ending and starting point – Multiply the differences by a common constant to convert them to the smallest possible integers u, v, w – The three indices are not separated by commas and are enclosed in square brackets: [uvw] – If any of the indices is negative, a bar is placed in top of that index
- 35. EXAMPLES: DIRECTIONS • Draw a [1,-1,0] direction within a cubic unit cell • Determine the indices for this direction – Answer: [120]
- 36. CRYSTALLOGRAPHIC PLANES • Crystallographic planes specified by 3 Miller indices as (hkl) • Procedure for determining h,k and l: – If plane passes through origin, translate plane or choose new origin – Determine intercepts of planes on each of the axes in terms of unit cell edge lengths (lattice parameters). Note: if plane has no intercept to an axis (i.e., it is parallel to that axis), intercept is infinity (½ ¼ ½) – Determine reciprocal of the three intercepts (2 4 2) – If necessary, multiply these three numbers by a common factor which converts all the reciprocals to small integers (1 2 1) – The three indices are not separated by commas and are enclosed in curved brackets: (hkl) (121) – If any of the indices is negative, a bar is placed in top of that index 1/2 1/2 1/4 (1 2 1) X Y Z
- 37. THREE IMPORTANT CRYSTAL PLANES ( 1 0 0) (1 1 1)(1 1 0)
- 38. THREE IMPORTANT CRYSTAL PLANES • Parallel planes are equivalent
- 39. EXAMPLE: CRYSTAL PLANES • Construct a (0,-1,1) plane
- 40. FCC & BCC CRYSTAL PLANES • Consider (110) plane • Atomic packing different in the two cases
- 41. Family of planes • All planes that are crystallographically equivalent— that is having the same atomic packing, indicated as {hkl} – For example, {100} includes (100), (010), (001) planes – {110} includes (110), (101), (011), etc.
- 42. Family of planes and directions • Miller indices of the specific plane (hkl). • Family of planes {hkl} • Crystallographic direction are indicated by [uvw] • Family of Crystallographic directions are indicated by <uvw>