L03B: Chapter 3 (continued) Note that an understanding of crystal structure is essential for doing well in the rest of this course. So you should be reading the text and doing example problems. Review the lectures and make certain you understand everything. If you don't, ask questions by In this lecture we cover the following: −Closed-packed metal structures: Face-centered cubic and hexagonal close packed. −Methods to denote directions and planes in hexagonal structures. VERY DIFFERENT ! −Polymorphism in carbon: diamond, graphite, graphene, buckeyballs, nano- fibers, amorphous, etc. W.R. Wilcox, Clarkson University. Last revised September 12, 2013

2 A sites B B B B B BB C sites C C C A B B ABCABC... Stacking sequence of {111} close-packed planes. FCC Stacking Sequence B B B B B BB B sites C C C A C C C A

ABAB... Stacking Sequence for close-packed planes in HCP APF = 0.74 Hexagonal Close-Packed Structure (HCP) 6 atoms/unit cell examples: Cd, Mg, Ti, Zn c/a = c a A sites B sites A sites Bottom layer Middle layer Top layer The only difference between FCC and HCP is second-nearest neighbors. Hexagonal unit cell

Crystallographic Directions in a Hexagonal Structure Miller-Bravais lattice 4 axes: a 1, a 2, a 3, z Dimensions are a (for a 1, a 2, and a 3 axes) and c (for z-axis) Direction [uvtw] Algorithm to draw vector. –Remove brackets –Divide by largest integer so all values are ≤ 1 –Multiply terms by appropriate unit cell dimension (a or c) to produce projections. –Construct vector by stepping off these projections.

5 Example of Drawing a Direction in a Hexagonal Lattice Draw the direction in a hexagonal unit cell. [1213] 4. Construct Vector 1. Remove brackets Algorithm a 1 a 2 a 3 z 2. Divide by 3 3. Projections proceed – a /3 units along a 1 axis to point p –2 a /3 units parallel to a 2 axis to point q a /3 units parallel to a 3 axis to point r c units parallel to z axis to point s p q r s start at point o [1213] direction represented by vector from point o to point s

1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of three- axis ( a 1, a 2, and z ) unit cell dimensions a and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas, for three-axis coordinates 5. Convert to four-axis Miller-Bravais lattice coordinates using equations below: 6. Adjust to smallest integer values and enclose in brackets [uvtw] Algorithm Determination of Miller-Bravais Indices for Direction

7 4. Brackets [110] 1. Reposition not needed 2. Projections a a 0 c Reduction Example a 1 a 2 z 5. Convert to 4-axis parameters 1/3, 1/3, -2/3, 0 => 1, 1, -2, 0 => [ 1120 ] 6. Reduction & Brackets Example Determination of Indices for Direction Determine indices for green vector

8 Denoting Crystallographic Planes in a Hexagonal Lattice example a 1 a 2 a 3 c 4. Miller-Bravais Indices(1011) 1. Intercepts 1  1 2. Reciprocals 1 1/  Reduction1 0 1 a2a2 a3a3 a1a1 z

Names of planes Three names are commonly used for crystallographic planes in the hexagonal system: basal, prismatic and pyramidal. For example, in ice: The basal plane is [0001]. Three prismatic planes are [1000], [0100] and [0010]. The pyramidal planes intersect the c axis at an angle. Example of a hexagonal pyramid:

Polymorphic Forms of Carbon Very strong covalent tetrahedral bonding. Consequently, very few free electrons and so is an electrical insulator. Single crystal diamond has many exceptional properties, e.g.: –Hardest material –Highest thermal conductivity Diamond cubic structure. Can also be considered face-centered cubic, but not close packed. Each fcc lattice site has 2 atoms. Diamond  VMSEVMSE The group IV semiconductors, Si and Ge, also have the diamond structure. Integrated circuits are made from Si. Hexagonal diamond (Lonsdaleite) discovered in meteorites:

Diamond synthesis Diamond is thermodynamically stable only at high pressure. Created in the earth at high pressure. Graphite is the stable structure at atmospheric conditions. At room temperature, the rate of transformation to graphite is negligible. Crystals, powder and coatings are made synthetically: High pressure Low pressure by forming H  and CH 3  with high T or plasma. e.g.: Many applications for lab-created diamond, e.g. hard coatings and abrasives.

Graphite  VMSEVMSE Layers with hexagonal structures. Very strong covalent bonding within each hexagonal layer. Very weak van der Waal’s bonding between layers. Very anisotropic properties. Good electrical conductor within layers. Easy separation of the layers. Comes in various forms, including small crystals. Has many applications. For example, see Hexagonal BN has the same structure, with alternating B & N atoms: Polymorphism for elements is called allotropy Compounds can also show polymorphism.

Graphene A very hot two-dimensional material. See, for example, Originally made by pulling adhesive tape from graphite crystals and dissolving the tape in a solvent. Very unusual thermal, mechanical, chemical, and electronic properties. Many potential applications have been demonstrated in the lab. The material of the future?

Carbon nanotubes Consists of a graphene sheet in the form of a seamless cylinder and closed by a cap on the end. A one-dimensional structure! May have a single wall (graphene layer) or multiple wall, and joined in different ways. Also very unusual properties and many potential applications.

Buckminsterfullerene Molecule “Buckey balls” C 60 molecule consisting of 20 hexagons and 12 pentagons, similar to a soccer ball. Covalent bonding. Unusual chemical properties. Possible use for hydrogen storage. For 3D view, open the following in Chrome: Three forms of amorphous carbon with commercial applications: Glassy, or vitreous, carbon: Carbon fibers Diamond-like carbon (DLC):