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1Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering1How do atoms arrange themselves to form solids?• Fundamental concepts and language• Unit cells• Crystal structures¾Simple cubic ¾Face-centered cubic¾ Body-centered cubic¾ Hexagonal close-packed• Close packed crystal structures• Density computations• Types of solidsSingle crystalPolycrystallineAmorphousChapter OutlineIntroduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering2Types of SolidsCrystalline material: atoms self-organize in a periodic arraySingle crystal: atoms are in a repeating or periodic array over the entire extent of the materialPolycrystalline material: comprised of many small crystals or grainsAmorphous: lacks a systematic atomic arrangementCrystalline AmorphousSiO22Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering3Crystal structureTo discuss crystalline structures it is useful to consider atoms as being hard spheres with well-defined radii. In this hard-sphere model, the shortest distance between two like atoms is one diameter.We can also consider crystalline structure as a lattice of points at atom/sphere centers.Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering4Unit CellThe unit cell is the smallest structural unit or building block that can describe the crystal structure. Repetition of the unit cell generates the entire crystal.Different choices of unit cells possible, generally choose parallelepiped unit cell with highest level of symmetryExample: 2D honeycomb net can be represented by translation of two adjacent atoms that form a unit cell for this 2D crystalline structureExample of 3D crystalline structure:3Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering5Metallic Crystal Structures¾ Metals are usually (poly)crystalline; although formation of amorphous metals is possible by rapid cooling¾ As we learned in Chapter 2, the atomic bonding in metals is non-directional ⇒ no restriction on numbers or positions of nearest-neighbor atoms ⇒ large number of nearest neighbors and dense atomic packing¾ Atom (hard sphere) radius, R, defined by ion core radius - typically 0.1 - 0.2 nm¾ The most common types of unit cells are the faced-centered cubic (FCC), the body-centered cubic (BCC) and the hexagonal close-packed (HCP). Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering6The 14 Bravais ;attices grouped into 7 lattice types.The restrictions on the lattice parameters (a,b,c) and the angles of the unit cell are listed for each. Whatare the most comon ?4Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering7Miller indices- h,k,l- for naming points in the crystal lattice. The origin has been arbitrarily selected as the bottom left-back corner of the unit cell.Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering8(a) the coordinates of the six face centersand the center of the cube, (b) the location of the point ¼, ¾, ¼ which isfound by starting at the origin and moving a distance ao/4 in the x-direction,then 3 ao/4 in the y-direction, and finallyao/4 in the z-direction.5Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering9Positions, Directions, and Planesx, y, z – u, v, wXYZLattice Positions and Directions:1) Always establish an origin2) Determine the coordinates of the lattice points of interest3) Translate the vector to the origin if required by drawing a parallel line or move the origin.4) Subtract the second point from the first: u2-u1,v2-v1,w2-w15) Clear fractions and reduce to lowest terms6) Write direction with square brackets [uvw]7) Negative directions get a hat.Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering10Indices of Planes:1) Identify the locations where the plane intercepts the x, y, z axes as the fractions of the unit cell edge lengths a, b, c.2) Infinity if the plane is parallel.3) Take the reciprocal of the intercepts. 4) Clear any fraction but do not reduce to lowest terms. 5) Example: 1/3,1/3,1/3 is (333) not (111)!!!6) Use parentheses to indicate planes (hkl) again with a hat over the negative indices.7) Families are indicated by {hkl}Remember Terminology:Defined coordinate system: x, y, zRespective unit cell edge lengths: a, b, c Direction: Denoted by [uvw]Family of direction(s): Denoted by: <uvw>Plane: Denoted by: (hkl)Family of Plane(s): Denoted by: {hkl}6Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering11•Directions are always perp. to their respective planes, i.e. [111] perp. (111) (for cubic systems)•Families of equivalent planes are equal with respect to symmetrical structures, they do not have to be parallel. Equivalent planes must be translated to the correct atomic positions in order to maintain the proper crystal symmetry. • Families of directions are equivalent in absolute magnitude.• (222) planes are parallel to the (111) planes but not equal.• Intercepts for the (222) planes are 1/2,1/2,1/2• Intercepts for the (333) planes are 1/3,1/3,1/3, remember this is inwhat we call “reciprocal space”. If you draw out the (333) plane it isparallel to the (111) plane but not equivalent.ab(110)0,0,0-b-a(-110)Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and Engineering12Representation of a series each of (a) (001), (b) (110), and (c) (111) crystallographic planes.7Introduction To Materials Science, Chapter 3, The structure of crystalline solidsUniversity of Tennessee, Dept. of Materials Science and


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