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Handout#3 - 2211 CRYSTAL STRUCTURE- Chapter 3 (atomic arrangement) Why study this? Ductile-brittle transition in metals. Crystalline # amorphous - transparency In gases, atoms have no order. If atoms bonded to each other but there is no repeating pattern (short range order). e.g. water, glasses (AMORPHOUS - non-crystalline) If atoms bonded together in a regular 3-D pattern they form a CRYSTAL - long range order - like wall paper pattern or brick wall.

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Handout#3 - 2212 Non dense, random packing Dense, regular packing Dense, regular-packed structures tend to have lower energy. ENERGY AND PACKING

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Handout#3 - 2213 atoms pack in periodic, 3D arrays typical of: Crystalline materials... -metals -many ceramics -some polymers atoms have no periodic packing occurs for: Noncrystalline materials... -complex structures -rapid cooling crystalline SiO 2 noncrystalline SiO 2 "Amorphous" = Noncrystalline MATERIALS AND PACKING

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Handout#3 - 2214 FOR SOLID MATERIALS: Most METALS (>99%) are CRYSTALLINE. CERAMICS are CRYSTALLINE except for GLASSES which are AMORPHOUS. POLYMERS (plastics) tend to be: either AMORPHOUS or a mixture of CRYSTALLINE + AMORPHOUS (known as Semi-crystalline)

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Handout#3 - 2215 CRYSTALS Different ways of arranging atoms in crystals. Assume atoms are hard spheres and pack like pool/snooker balls (touching). Each type of atom has a preferred arrangement depending on Temp. and Pressure (most stable configuration). These patterns known as SPACE LATTICES

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Handout#3 - 2216 tend to be densely packed. have several reasons for dense packing: -Typically, only one element is present, so all atomic radii are the same. -Metallic bonding is not directional. -Nearest neighbor distances tend to be small in order to lower bond energy. have the simplest crystal structures. METALLIC CRYSTALS

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Handout#3 - 2217 7 types of CRYSTAL SYSTEM 14 standard UNIT CELLS METALLIC CRYSTAL STRUCTURES Most metals crystallize into one of three densely packed structures: BODY CENTERED CUBIC - BCC FACE CENTERED CUBIC - FCC HEXAGONAL (CLOSE PACKED) - HCP Actual size of UNIT CELLS is VERY VERY SMALL!! Iron unit cell length (0.287 x 10 -9 m) (0.287 nm) 1 mm length of iron crystal has 3.5 million unit cells

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Handout#3 - 2218 Rare due to poor packing (only Po has this structure) Close-packed directions are cube edges. Coordination # = 6 (# nearest neighbors) SIMPLE CUBIC STRUCTURE (SC)

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Handout#3 - 2219 APF for a simple cubic structure = 0.52 ATOMIC PACKING FACTOR

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Handout#3 - 22110 Coordination # = 8 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)

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Handout#3 - 22111 BCC STRUCTURE Atoms at cube corners and one in cube centre. Lattice Constant for BCC: e.g.Fe (BCC) a = 0.287 nm Two atoms in Unit Cell. (1 x 1 (centre)) + (8 x 1/8 (corners)) = 2 Each atom in BCC is surrounded by 8 others. COORDINATION number of 8. Packing is not as good as FCC; APF = 0.68 BCC metals include: Iron (RT), Chromium, Tungsten, Vanadium

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Handout#3 - 22112

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Handout#3 - 22113 APF for a body-centered cubic structure = 0.68 ATOMIC PACKING FACTOR: BCC

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Handout#3 - 22114 Coordination # = 12 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)

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Handout#3 - 22115 FACE CENTERED CUBIC (FCC) e.g. copper, aluminium, gold, silver, lead, nickel Lattice constant (length of cube side in FCC) “a” for FCC structure: where R = atomic radius Each type of metal crystal structure has its own lattice constant. (1/8 at each corner x 8) + (½ at each face x 6 ) = 4 So 4 atoms per Unit Cell. Each atom touches 12 others. Co-ordination number = 12.

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Handout#3 - 22116

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Handout#3 - 22117 APF for a body-centered cubic structure = 0.74 ATOMIC PACKING FACTOR: FCC

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Handout#3 - 22118 ABCABC... Stacking Sequence 2D Projection FCC Unit Cell FCC STACKING SEQUENCE

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Handout#3 - 22119 Coordination # = 12 ABAB... Stacking Sequence APF = 0.74 3D Projection 2D Projection Adapted from Fig. 3.3, Callister 6e. HEXAGONAL CLOSE-PACKED STRUCTURE (HCP)

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Handout#3 - 22120 HEXAGONAL CLOSE PACKED Note: not simple hexagonal but HCP Simple Hex. very inefficient; HCP has extra plane of atoms in middle. 1/6 of atom at each corner. So (1/6) x 12 corners=2 atoms and (½) x (top + bottom) =1 atom and (3) internal =3 atoms Total=6 atoms/cell Because of Hexagonal arrangement (not cubic), have 2 lattice parameters “a”, and “c”

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Handout#3 - 22121

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Handout#3 - 22122 a = basal side= 2R c = cell height By geometry, for IDEAL HCP: but this varies slightly for some HCP Metals. HCP metals include: Magnesium, Zinc, Titanium, Zirconium, Cobalt. atomic packing factor for HCP = 0.74(same as FCC) Atoms are packed as tightly as possible. Each atom surrounded by 12 other atoms so co- ordination number = 12.

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Handout#3 - 22123 CRYSTAL DENSITY The true density, , of material (free from defects) can be calculated knowing its crystal structure. n = number of atoms in unit cell A = Atomic Weight of element (g/mol) V c = volume of unit cell N av = Avogadro’s number (6.023 x 10 23 atoms/mol)

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Handout#3 - 22124 e.g., copper, FCC 4 atoms/cell n = 4 Cu atoms have mass 63.5 g/mol Vol. of cell = a 3, for FCC a =2R 2 Atomic radius of copper = 0.128 nm = 8.89 Mgm -3 (or 8.89 gcm -3 or 8890 kgm -3 )

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Handout#3 - 22125 POLYMORPHISM / ALLOTROPY Some elements/compounds can exist in more than one crystal form. Usually requires change in temperature or pressure. Carbon: Diamond (high pressure) or Graphite (low). Can be IMPORTANT as some crystal structures more dense (better packing, higher APF) than others, so a change in crystal structure can often result in volume change of material. APF e.g. Iron 913 o C FCC0.74 911 o C BCC 0.68 i.e. expands on cooling!

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Handout#3 - 22126 Why? Metals have... close-packing (metallic bonding) large atomic mass Ceramics have... less dense packing (covalent bonding) often lighter elements Polymers have... poor packing (often amorphous) lighter elements (C,H,O) Composites have... intermediate values Data from Table B1, Callister 6e. DENSITIES OF MATERIAL CLASSES

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Handout#3 - 22127 CRYSTAL SYSTEMS Group crystals depending on shape of Unit Cell. x, y and z are three axes of lattice separated by angles , and . A unit cell will have sides of length a, b and c. (Note: for the cubic system all sides equal so a = b = c) SEVEN possible crystal systems (Table 3.2) Cubicmost symmetry … Triclinic least symmetry

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Handout#3 - 22128

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Handout#3 - 22129

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Handout#3 - 22130 Positions in lattice

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Handout#3 - 22131 CRYSTALLOGRAPHIC DIRECTIONS Line between two points or vector. Using 3 coordinate axes, x, y, and z. Position vector so that it passes through origin (parallel vectors can be translated). Length of vector projected onto the three axes (x, y and z) is determined in terms of unit cell dimensions (a, b and c). Multiply or divide by common factor to reduce to lowest common integers. Enclose in SQUARE brackets with no commas [uvw], and minus numbers given by bar over number; e.g.

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Handout#3 - 22132

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Handout#3 - 22133 Parallel vectors have same indices. Changing sign of all indices gives opposite direction. If directions are similar, (i.e., same atomic arrangements - for example, the edges of a BCC cube) they belong to a FAMILY of directions: i.e. with brackets can change order and sign of integers. e.g. cube internal diagonals cube face diagonals

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Handout#3 - 22134 HEXAGONAL CRYSTALS Use a 4-axis system (Miller-Bravais). a 1, a 2 and a 3 axes in basal plane at 120 to each other and z axis in vertical direction. Directions given by [uvtw] or [a 1 a 2 a 3 c] Can convert from three-index to four index system. t=-(u+v)

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Handout#3 - 22135

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Handout#3 - 22136 CRYSTAL PLANES Planes specified by Miller Indices (hkl) (Reciprocal Lattice). Used to describe a plane (or surface) in a crystal e.g., plane of maximum packing. Any two planes parallel to each other are equivalent and have identical Miller indices

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Handout#3 - 22137

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Handout#3 - 22138

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Handout#3 - 22139

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Handout#3 - 22140 To find Miller Indices of a plane: If the plane passes through the selected origin, construct a parallel plan in the unit cell or select an origin in another unit cell. Determine where plane intercepts axes. (if no intercept i.e.., plane is parallel to axis, then ) e.g., axis xyz interceptabc Take reciprocals of intercepts (assume reciprocal of is 0): 1/a1/b1/c Multiply or divide to clear fractions: (hkl) Miller indices of plane

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Handout#3 - 22141 FAMILY of planes, use {hkl} These planes are crystallographically similar (same atomic arrangements). e.g., for cube faces: {100} NOTE:In CUBIC system only, directions are perpendicular to planes with same indices. e.g., [111] direction is perpendicular to the (111) plane. HEXAGONAL CRYSTALS Four-index system similar to directions; (hkil) i = - (h+K)

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Handout#3 - 22142 ATOMIC PACKING Arrangement of atoms on different planes and in different directions. LINEAR ATOMIC DENSITIES Tells us how well packed atoms are in a given direction. If LD = 1 then atoms are touching each other.

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Handout#3 - 22143 PLANAR DENSITIES Tells us how well packed atoms are on a given plane. Similar to linear densities but on a plane rather than just a line. gives fraction of area covered by atoms.

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Handout#3 - 22144 e.g., BCC unit cell, (110) plane: 2 whole atoms on plane in unit cell. SoA c = 2( R 2 ) AD = a, DE = a 2 And so A p = a 2 2

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Handout#3 - 22145 PACKING ON PLANES FCC and HCP are both CLOSE-PACKED structures. APF = 0.74 (This is the maximum if all atoms are same size). Atoms are packed in CLOSE-PACKED planes In FCC, {111} are close packed planes In HCP, (0001) is close packed Both made of close packed planes, but different stacking sequence. FCC planes stack as ABCABCABC HCP planes stack as ABABABABAB BCC is not close packed (APF = 0.68) most densely packed plane is {110}

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Handout#3 - 22146

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Handout#3 - 22147

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Handout#3 - 22148

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Handout#3 - 22149 Some engineering applications require single crystals: Crystal properties reveal features of atomic structure. --Ex: Certain crystal planes in quartz fracture more easily than others. diamond single crystals for abrasives --turbine blades CRYSTALS AS BUILDING BLOCKS

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Handout#3 - 22150 SINGLE CRYSTALS This is when a piece of material is made up of one crystal; all the unit cells are aligned up in the same orientation. POLYCRYSTAL Many small crystals (grains) with different orientations joined together. Most materials/metals are POLYCRYSTALLINE. Grain boundary - Regions where grains (crystals) meet.

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Handout#3 - 22151 Most engineering materials are polycrystals. Nb-Hf-W plate with an electron beam weld. Each "grain" is a single crystal. If crystals are randomly oriented, overall component properties are not directional. Crystal sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers). 1 mm POLYCRYSTALS

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Handout#3 - 22152

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Handout#3 - 22153 ANISOTROPY Many properties depend on direction in crystal in which they are measured. E.g. Stiffness (rigidity) electrical conductivity, refraction. If property varies with direction - Anisotropic. If no variation with direction - Isotropic Single crystals show this variation. Polycrystalline materials are usually randomly oriented so effect is evened out to give average values in all directions.

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Handout#3 - 22154 Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (E poly iron = 210 GPa) -If grains are textured, anisotropic. 200 m SINGLE VS POLYCRYSTALS

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Handout#3 - 22155

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Handout#3 - 22156 FINDING OUT WHAT THE CRYSTAL STRUCTURE IS. Analysed using X-Rays. X-Ray Diffraction. X-rays are diffracted off atoms and either constructively interfere (peak) or destructively interfere (low) from layers of atoms depending on interplanar spacing (d hkl ) and angle. n = 2d hkl sin (Bragg's Law) n = 1,2, 3, 4, 5....... = wavelength of incident X-rays = incident angle

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Handout#3 - 22157

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Handout#3 - 22158

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Handout#3 - 22159 Incoming X-rays diffract from crystal planes. Measurement of: Critical angles, c, for X-rays provide atomic spacing, d. X-RAYS TO CONFIRM CRYSTAL STRUCTURE

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Handout#3 - 22160

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Handout#3 - 22161 So can measure peak and determine d hkl and then “a”. Distance between similar planes in the cubic systems, e.g., (110) planes in adjacent unit cells:

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Handout#3 - 22162 NON-CRYSTALLINE SOLIDS Non crystalline solids are amorphous materials.i.e.. they are not crystalline. They have no long range order. Short range order only. Structure is usually too complex to form crystals when cooled from liquid at normal rates. E.g.. Glasses, some plastics,

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Handout#3 - 22163

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Handout#3 - 22164 Atoms may assemble into crystalline or amorphous structures. We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP). Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but properties are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains. SUMMARY

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