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 email (wilcox@clarkson.edu).• 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

BB

B B

C sites

C C

CA

B

B sites

ABCABC... Stacking sequence of {111} close-packed planes.

FCC Stacking Sequence

B B

B

BB

B B

B sites

C C

CA

C C

CA

AB

C

ABAB... Stacking Sequence for close-packed planes in HCP

APF = 0.74

Hexagonal Close-Packed Structure (HCP)

6 atoms/unit cellexamples: Cd, Mg, Ti, Zn

c/a = 1.633

c

a

A sites

B sites

A sitesBottom 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: a1, a2, a3, z

• Dimensions are a (for a1, a2, and a3 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 -1 -2 1 3

Algorithm a1 a2 a3 z

2. Divide by 3

3. Projections

proceed –a/3 units along a1 axis to point p

–2a/3 units parallel to a2 axis to point q

a/3 units parallel to a3 axis to point r

c units parallel to z axis to point s

p

qr

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 (a1, a2, and z) unit cell dimensions a and c 3. Adjust to smallest integer values4. 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

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4. Brackets [110]

1. Reposition not needed

2. Projections a a 0c1 1 0

3. Reduction 1 1 0

Example a1 a2 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 a1 a2 a3 c

4. Miller-Bravais Indices (1011)

1. Intercepts 1 -1 12. Reciprocals 1 1/

1 0 -1-1

11

3. Reduction 1 0 -1 1

a2

a3

a1

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

VMSE

• The group IV semiconductors, Si and Ge, also have the diamond structure.

• Integrated circuits are made from Si.• Hexagonal diamond (Lonsdaleite)

discovered in meteorites:http://en.wikipedia.org/wiki/Lonsdaleite

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 CH3 with high T or plasma.• e.g.: http://people.clarkson.edu/~lregel/actaastr.pdf

• Many applications for lab-created diamond, e.g. hard coatings and abrasives.

Graphite

VMSE

• 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 https://en.wikipedia.org/wiki/Graphite

• Hexagonal BN has the same structure, with alternating B & N atoms:http://en.wikipedia.org/wiki/Boron_nitride

• Polymorphism for elements is called allotropy

• Compounds can also show polymorphism.

Graphene

• A very hot two-dimensional material. See, for example, http://en.wikipedia.org/wiki/Graphene .

• 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.• http://en.wikipedia.org/wiki/Carbon_nanotube

Buckminsterfullerene Molecule

• “Buckey balls”

• C60 molecule consisting of 20 hexagons and 12 pentagons, similar to a soccer ball.

• Covalent bonding.• Unusual chemical properties.• Possible use for hydrogen storage.• http://en.wikipedia.org/wiki/Buckminsterfullerene

• For 3D view, open the following in Chrome:http://www.3dchem.com/molecules.asp?ID=217#

• Three forms of amorphous carbon with commercial applications:• Glassy, or vitreous, carbon:

http://en.wikipedia.org/wiki/Glassy_carbon • Carbon fibers

http://en.wikipedia.org/wiki/Carbon_(fiber) • Diamond-like carbon (DLC):

http://en.wikipedia.org/wiki/Diamond-like_carbon