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Unformatted text preview: Carbon nanostructures
(http://www.mf.mpg.de/de/abteilungen/schuetz/index.php?lang=en&content=researchtopics&type=specific&name=h2storage) 1 • Crystalline Material: atoms are situated in a periodic array over large atomic distances repetitive three dimensional pattern/ arrangement of atoms– lattice structure .
– repeat unit called unit cell – repeated pattern called crystal lattice • • • Non‐crystalline / Amorphous Material: do not crystallize and absence of long‐rang atomic order Crystal Structure: manner in which atoms, ions or molecules are spatially arranged Many possible structures defined by
– shape of cell – arrangement of atoms within cell 2 • •
Crystalline metals
usually e.g. steel, brass often e.g. alumina never “crystalline” polymers always partly amorphous Amorphous
rarely e.g. metallic glass often e.g. soda glass Mixed
never ceramics often e.g. silicon nitride polymers usually e.g. polyethylene sometimes e.g. nylon 3 • • • 4
http://www.tutorvista.com/content/chemistry/chemistry‐iv/solid‐state/space‐lattice.php •
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e.g. Dielectric constant capacitance •
Strength Electrical conductivity
5 • •
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6 Square packing: Each circle occupies an 'equivalent area' of 4r2, because no other sphere can use this area. Hexagonal packing: Each circle occupies a smaller 'equivalent area', making this a more efficient packing system. This is close packing in 2 dimensions. J. Hiscocks, 2003 J. Hiscocks, 2003 Area 2r 2 4r2 Area √3/2 2r 2 3.464 r2
7 See http://www.drking.worldonline.co.uk/hexagons/misc/area.html • The curve of the watchglass pushes the spheres together.
– equivalent to a bonding force • 2‐D close packing occupies the smallest area and lowers the overall energy.
– any spheres that achieve hexagonal packing will stay that way • Any other arrangement e.g. square array is unstable.
J. Hiscocks, 2003 8 9 Simple Cubic • •
– • • 2R a
10 Body‐Centered Cubic • α‐Fe, Cr, W, Mo transition metals • atoms sit at
• – cell corners – cell center Number of Atoms/ Unit Cell: 2 4 a R 3
APF: 0.68 CN: 8
Courtesy P. M. Anderson 11 • Cu, Al, Ag, Au, γ‐Fe • Atoms sit at – cell corners – middle of cell face Face‐Centered Cubic • Co‐ordination number CN 12 • Counting up atoms
– How many neighbouring cells share each atom? – 4 atoms/cell • Atomic packing factor APF 0.74 4 R 2a a 2 2R
12 • FCC and HCP crystals are both based on close‐packed planes. • FCC ABCABC… sequence • HCP ABABAB… sequence • For both CN 12 APF 0.74 • Ideally, the HCP c/a 1.633 but it often deviates from this. 13 •
– •
– – • •
14 3types: SC, BCC, FCC 1type: HCP 2types: 1type: 4types: 2types: 1type:
15 16 Crystal Structure Structure Hard Sphere Model Simple Cubic Reduced Sphere Unit Cell Details Number of Atoms/ Unit Cell Atomic Packing Factor Coordination Number a fR 1
Body Centered Cubic ‐‐‐‐ 6 2 0.68 8 Face Centered Cubic 4 0.74 12 Hexagonal Close Packed 6 ‐‐‐‐ 0.74 12
17 •
– –
• cell volume, Vc m3 • No. of atoms/cell, n n A Vc N A number
Atoms/volume NA Avogadro’s Mass/atom 18 α
• Which has the higher density? A / NA is the same for each r ~ n/a3 BCC FCC β ~
a 2 a3
4 R 3 ~ 4 a3 a 2 2R ~ 23 3 1 64 R 3 0.16 R3 ~ 41 16 2 R 3 0.177 R3
19 ~ ~ •
– – –
o o o o o • 20 •
– • e.g. light bulb filaments •
– – http://science.howstuffworks.com/fluorescentlamp1.htm 21 • Heat wire until it glows red.
– It is now γ ‐ FCC iron. – Cool wire to room temperature. Watch how the length of the wire changes during cooling. http://matse1.mse.uiuc.edu/~tw/metals/d.html 22 23 • Need a nomenclature to describe crystal structures in detail. • In particular:
– directions – planes within crystals • The method should be independent of cell type. Can’t use Cartesian co‐ordinates. 24 1. 2. 3. Start at any cell corner. Find coordinates of vector in units of a, b, c. Multiply or divide all the coordinates by a common factor.
– To reduce all the coordinates to the smallest possible integer values. 4. 5. Represent as u v w – no commas Represent negative directions as ū. This is called the Miller Index for direction. 25 • What is the Miller index for A? a 2 0 1 b 0 2 1 c 1 1 2 d other • What is the Miller index of B? a 2 2 1 x b 1 1 2 c 1 2 1 d other z B A y 26 • The Miller index of a plane is the same as the Miller index of the direction normal to the plane.
– Choose a starting point origin so that the plane does not pass through the origin. – Find the intercepts in units of x, y, z, planes parallel to an axis have an intercept at . – Find the reciprocals of the intercepts: 1/x, 1/y, 1/z. – Multiply or divide by common factor to get the smallest possible integer values. – Represent the index as h k l – no commas. – Represent negative values using the bar: h
27 What is the Miller index of this plane? Intercepts: a b ‐1 c 1/2 0 –1 2 not needed 0 1 2 Reciprocals: Reduction: Index: 28 29 z z y y x x (110) ( 2 11) Identify the Miller indices of these planes
30 Eg. 1 Eg. 2 Draw the line 321 ! 1/3
Find the Miller indices of this line! [1 1 1] 2/3 31 • 1 0 0 direction has 5 cousins:
[001] [100] [010], [001], [1 00], [0 1 0], [00 1] _ _ _ [010] call this the 1 0 0 family [010] [100] [001] The three most important families of directions are: 100 , 110 , 111
32 • (111) (111) (111) 33 •
– – • 34 35 • Single Crystals: – Grow large pieces of material with the same crystal orientation – repeated arrangement of atoms throughout the entity of the specimen – Rare in nature, difficult to grow. – Gem Stone – Si wafers, quartz oscillators. • Examples: • Most materials contain many crystals called grains polycrystal / polycrystalline. • The region of atomic mismatch where grains meet is called a grain boundary atomic dimensions . 36 • Often, the physical properties of a material differ depending on the crystallographic direction in which the measurement is taken, e.g. conductivity, elastic modulus, index of refraction
Fuchsite Mica • Isotropic: Substances in which measured properties are independent of the direction of measurement are referred to as isotropic. • structural symmetry decreases anisotropy increases • Highly anisotropic crystals include:
– graphite hexagonal with a large c/a value . – mica sheet silicate . E (diagonal) = 273 GPa E (edge) = 125 GPa
BCC Fe
37 • X‐Rays help determine atomic interplanar distances and crystal structures • A form of electromagnetic Radiation with high energy and short wavelengths • Diffraction occurs when a wave encounters a series of regularly spaced obstacles that:
– Are capable of scattering the wave – Have spacings that are comparable in magnitude to the wavelength • Diffraction: Constructive Interference of x‐ray beams that are scattered by atoms of a crystal. • When two scattered waves are: – In Phase Constructive Interference – Out of Phase Destructive Interference
38 • • Crystals diffract X‐rays Bragg’s law says constructive interference will occur if the extra path is a multiple of the wavelength:
n=2dsin Note: For practical reasons, the “diffraction angle” is 2 39 • From Bragg’s Law: n 2 d sin d spacing between the planes • We can show that for any h k l plane: a d h2 k 2 l 2
40 •
– 3 4 23 • θ
41 • Also called amorphous solids or glass. • Caused by irregular arrangements of the molecular units.
– eg. SiO44‐ tetrahedra in window glass • Amorphous solids show short‐range order, but not long‐range order.
– no X‐ray diffraction patterns
42 • • Free energy G glass liquid solid – – glass solid
T •
–
2
2 43 • • • 44 • • • • • •
45 3 12] b) [211] and [211] c) [201] and [111] d) [002] and [211] ...
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