Carbon nanostructures. (
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1 Carbon nanostructures ( 1
2 Crystal Structures Crystalline Material: atoms arrange into a periodic array (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, i.e. there is no long-range 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
3 Atomic Packing in Solids regular atomic packing crystalline random atomic packing amorphous, glassy Crystalline Amorphous Mixed metals usually (e.g. steel, brass) rarely (e.g. metallic glass) never ceramics often (e.g. alumina) often (e.g. soda glass) often (e.g. silicon nitride) polymers never ( crystalline polymers always partly amorphous) usually (e.g. polyethylene) sometimes (e.g. nylon) 3
4 Unit Cells Basic structural unit or building block of crystal structure Represents the symmetry of the crystal structure Defines geometry and atom positions 4
5 Properties / Crystal Structure Relationship Many material properties are influenced by the crystal structure including: e.g. Dielectric constant (capacitance) Strength Ductility Electrical conductivity 5
6 Metallic Crystals Metallic bonding non-directional tends to form cubic, close-packed structures 3 main variants face-centered cubic (FCC) body-centered cubic (BCC) hexagonal close packed (HCP) 6
7 Close Packing in Two Dimensions Square packing: Each circle occupies an 'equivalent area' of 4r 2, 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 = 4r 2 Area = ( 3/2)(2r) 2 = r 2 See 7
8 Demo: Close Packing in Two Dimensions 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,
9 Metallic Crystal Structures 9
10 Packing atoms in Three Dimensions Most metals have one of the following unit cells: face-centered cubic (FCC) body-centered cubic (BCC) hexagonal close packed (HCP) 10
11 SC Crystals (Simple Cubic) Not very common. Atoms sit at cell corners 1 atom/cell Atomic Hard sphere model 2R a 11
12 SC Crystals Simple Cubic Crystal Structure 12
13 α-fe, Cr, W, Mo (transition metals) atoms sit at cell corners cell center Number of Atoms/ Unit Cell: 2 a 4 3 R BCC Crystals (Body-Centered Cubic) APF: 0.68 (a portion of the unit cell that is occupied by atoms) CN: 8 Courtesy P. M. Anderson 13
14 BCC Crystals Body-Centered Cubic Crystal Structure (Click to Play) 14
15 FCC Crystals (Face-Centered Cubic) Cu, Al, Ag, Au, γ-fe Atoms sit at cell corners middle of cell face Co-ordination number (CN) = 12 Counting up atoms How many neighbouring cells share each atom? 4 atoms/cell Atomic packing factor (APF) = R 2a a 2 2R 15
16 Crystal Structure Closed packed lattices FCC: ABCABC layers HCP: ABABAB layers
17 FCC Crystal Structure This image is the property of IBM Corporation. Courtesy P. M. Anderson Scanning tunnelling microscope image of a Ni surface. 17
18 FCC Crystals 18
19 FCC Crystals Face-Centered Cubic Crystal Structure (Click to Play) 19
20 HCP Crystals 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 = but it often deviates from this. 20
21 Crystal Structure Closed packed lattices FCC: ABCABC layers HCP: ABABAB layers
22 Close Packed Structure FCC and HCP are close packed structures. BCC is not. consider the FCC <111> plane: 6-fold coordination of the A-layer. two sets of positions for the next layer, B or C. FCC uses ABCABC stacking. Whereas, HCP used ABABAB 22
23 HCP Crystals (Click to Play) (Click to Play) Hexagonal Close-Packed Crystal Structure 23
24 Other Structures 3-types: SC, BCC, FCC 1-type: HCP 2-types: 1-type: 4-types: 2-types: 1-type: 24
25 Crystal Structures in the Periodic Table 25
26 Crystal Structure Details Structure (Hard Sphere Model) Reduced Sphere Unit Cell Number of Atoms/ Unit Cell a=f(r) Atomic Packing Factor Coordination Number Simple Cubic Body Centered Cubic Face Centered Cubic Hexagonal Close Packed 6 a=2r C=1.63a
27 Density Density is a function of: atomic weight, A (g/mol) crystal structure cell volume, V c (m 3 ) No. of atoms/cell, n number Atoms/volume n V A c N A Mass/atom N A =Avogadro s 27
28 Polymorphism Fe is polymorphic (has more than one crystal structure) α - BCC at T < 910 o C γ - FCC at 910 o C < T < 1394 o C δ - BCC at 1394 o C < T <1538 o C Does anyone know what happened to β- Fe? 28
29 ALS: Density Calculation Fe has two forms α (BCC) and (FCC) Which has the higher density? Hint: Keep in mind that A / N A is the same for both structures BCC FCC 2 ~ a 4 ~ a 3 3 a 4 3 R R a 2 2R 4 1 ~ 16 2 R ~ ~ ~ R 3 R 3 29
30 Crystallographic Directions and Planes 30
31 Miller Indices 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. 31
32 Miller Direction Indices 1. Start at any cell corner. 2. Find coordinates of vector in units of a, b, c. 3. Multiply or divide all the coordinates by a common factor. To reduce all the coordinates to the smallest possible integer values. 4. Represent as [ u v w ] no commas 5. Represent negative directions as ū. This is called the Miller Index for direction. 32
![Miller Direction Indices 1. Head point coordinate, a b c. 2. Tail point coordinate, k l m. 3. a k= u, b l=v, c-m=w 4. Represent as [ u v w ] no commas 5.](/docs-images/77/76439562/images/33-0.jpg "Multiply or divide all the coordinates by a common factor. - To reduce all the coordinates to the smallest possible integer values. 6. Represent negative directions as ū.")
33 Miller Direction Indices 1. Head point coordinate, a b c. 2. Tail point coordinate, k l m. 3. a k= u, b l=v, c-m=w 4. Represent as [ u v w ] no commas 5. Multiply or divide all the coordinates by a common factor. - To reduce all the coordinates to the smallest possible integer values. 6. Represent negative directions as ū. This is called the Miller Index for direction. 33
34 ALS: Miller Index for Direction What is the Miller index for A? a) [ ] b) [ ] c) [ ] d) other B z A What is the Miller index of B? a) [ ] b) [ ] c) [ ] d) other x y 34
35 Miller Indices for Planes 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 35
36 ALS: Miller Index of a Plane What is the Miller index of this plane? Intercepts: Reciprocals: Reduction: Index: ( ) a = b = -1 c = 1/2 not needed 36
37 Common Miller Indices 37
38 z Examples z y y x x (110) Identify the Miller indices of (211 these ) planes 38
39 Examples Eg. 1 Eg. 2 Draw the line [321]! Find the Miller indices of this line! [111] 1 2/3 1/3 39
40 Families of Directions [ ] direction has 5 cousins: _ [010], [001], [100], [010], [001] [001] [100] [010] call this the < > family [010] [100] [001] The three most important families of directions are: <100>, <110>, <111> 40
41 Family of Planes The (111) plane also has many cousins: e.g. (111) (111) (111) Call this the {111} family. 41
42 How are Lines and Planes Related? How can we tell if the [ u v w ] direction lies in the ( h k l ) plane? recall: The ( h k l ) represents the vector normal to the plane. recall: The dot product between normal vectors is zero. You can treat Miller indices like ordinary vectors: [ u v w ] lies in ( h k l ) if ( h k l ) ( u v w ) = h u + k v + l w = 0 42
43 Additional Concepts 43
44 Discovery of X-Rays Wilhelm Conrad Röntgen Rontgen's first x ray image. The ring can be seen on his wife's hand (1895). Won 1 st Nobel Prize in physics (1901).
45 X-Ray
46 X-Ray Diffraction 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 46
47 How Do We Know It s A Crystal? 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 47
48 Miller Indices and Planar Spacing From Bragg s Law: n = 2 d sin We can show that for any ( h k l ) plane: d = spacing between the planes d h 2 a k 2 l 2 48
49 Diffraction from a Crystalline Solid SiAlON is a Si 3 N 4 -Al 2 O 3 alloy Used for cutting tools (very hard) 2 phases present (a-cubic, b-hexagonal) θ 49
50 Polycrystals Single Crystals: Some materials consist of one crystal. Rare in nature, difficult to grow. Examples: Gem Stone Si wafers, quartz oscillators. Most materials contain many crystals called grains polycrystal / polycrystalline. The region of atomic mismatch where grains meet is called a grain boundary (atomic dimensions). 50
51 Anisotropy Often, the physical properties of a material differ depending on the crystallographic direction in which the measurement is taken anisotropy e.g. conductivity, elastic modulus, index of refraction Fuchsite Mica Isotropic: Properties which are independent of the direction of measurement are referred to as being isotropic. As structural symmetry decreases, anisotropy increases Highly anisotropic crystals include: graphite (hexagonal with a large c/a value). mica (sheet silicate). BCC Fe 51
52 Non-Crystalline Solids Also called amorphous solids or glass. Caused by irregular arrangements of the molecular units. eg. SiO 4 4- tetrahedra Amorphous solids show short-range order, but not longrange order. no X-ray diffraction patterns 52
53 Silica Glass Glass is a supercooled liquid. Free energy G solid glass liquid G glass > G solid Glass is unstable and therefore hard to make. T Network modifiers help (CaO, Na 2 O). They break up the SiO 2 network. 53
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