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UVA MSE 2090 - Chapter Outline

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1MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsHow do atoms arrange themselves to form solids?• Fundamental concepts and language• Unit cells• Crystal structures¾ Face-centered cubic¾ Body-centered cubic¾ Hexagonal close-packed• Close packed crystal structures• Density computations• Types of solidsSingle crystalPolycrystallineAmorphous3.8–3.10 Crystallography – Not Covered / Not Tested3.15 Anisotropy – Not Covered / Not Tested3.16 Diffraction – Not Covered / Not TestedChapter Outline2MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsTypes 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: disordered – lack of a systematic atomic arrangementCrystalline AmorphousSiO23MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsCrystal structuresWhy do atoms assemble into ordered structures (crystals)?Energy of interatomic bond0Interatomic distanceLet’s consider nondirectionalbonding (like in metals)Energy of the crystal < Energy of the amorphous solid4MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsCrystal 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 of the hard sphere.We can also consider crystalline structure as a lattice of points at atom/sphere centers.2R- hard-sphere model5MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsUnit CellThe unit cell is a 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, we will consider 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:6MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsMetallic 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 Atomic (hard sphere) radius, R, defined by ion core radius - typically 0.1 - 0.2 nm¾ The most common types of unit cells are• faced-centered cubic (FCC)• body-centered cubic (BCC)• hexagonal close-packed (HCP).7MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsFace-Centered Cubic (FCC) Crystal Structure (I)¾ Atoms are located at each of the corners and on the centers of all the faces of cubic unit cell¾ Cu, Al, Ag, Au have this crystal structureTwo representations of the FCC unit cell8MSE 2090: Introduction to Materials Science Chapter 3, Structure of solids¾ The hard spheres touch one another across a face diagonal ⇒ the cube edge length, a= 2R√2¾ The coordination number, CN = the number of closest neighbors to which an atom is bonded = number of touching atoms, CN = 12¾ Number of atoms per unit cell, n = 4. (For an atom that is shared with m adjacent unit cells, we only count a fraction of the atom, 1/m). In FCC unit cell we have:6 face atoms shared by two cells: 6×1/2 = 38 corner atoms shared by eight cells: 8×1/8 = 1¾ Atomic packing factor, APF = fraction of volume occupied by hard spheres = (Sum of atomic volumes)/(Volume of cell) = 0.74 (maximum possible)Face-Centered Cubic Crystal Structure (II)Ra9MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsmaximum possible packing of hard spheresFace-Centered Cubic Crystal Structure (III)RaAPF = (Sum of atomic volumes)/(Volume of unit cell)Let’s calculate the atomic packing factor for FCC crystalVolume of 4 hard spheres in the unit cell:3344 Rπ×22Ra =Volume of the unit cell:21633Ra =74.02321631633=π=π= RRAPF10MSE 2090: Introduction to Materials Science Chapter 3, Structure of solids¾ Corner and face atoms in the unit cell are equivalent¾ FCC crystal has APF of 0.74, the maximum packing for a system equal-sized spheres ⇒ FCC is a close-packed structure¾ FCC can be represented by a stack of close-packed planes (planes with highest density of atoms)Face-Centered Cubic Crystal Structure (IV)11MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsBody-Centered Cubic (BCC) Crystal Structure (I)Atom at each corner and at center of cubic unit cellCr, α-Fe, Mo have this crystal structure12MSE 2090: Introduction to Materials Science Chapter 3, Structure of solids¾ The hard spheres touch one another along cube diagonal ⇒ the cube edge length, a= 4R/√3¾ The coordination number, CN = 8¾ Number of atoms per unit cell, n = 2Center atom (1) shared by no other cells: 1 x 1 = 18 corner atoms shared by eight cells: 8 x 1/8 = 1¾ Atomic packing factor, APF = 0.68¾ Corner and center atoms are equivalentBody-Centered Cubic Crystal Structure (II)a13MSE 2090: Introduction to Materials Science Chapter 3, Structure of solidsHexagonal Close-Packed Crystal Structure (I)¾ HCP is one more common structure of metallic crystals ¾ Six atoms form regular hexagon, surrounding one atom in center. Another plane is situated halfway up unit cell (c-axis), with 3 additional atoms situated at interstices of hexagonal (close-packed) planes¾ Cd, Mg, Zn, Ti have this crystal structure14MSE 2090: Introduction to Materials Science Chapter 3, Structure of solids¾ Unit cell has two lattice parameters a and c. Ideal ratio c/a = 1.633¾ The coordination number, CN = 12 (same as in FCC)¾ Number of atoms per unit cell, n = 6. 3 mid-plane atoms shared by no other cells: 3 x 1 = 312 hexagonal corner atoms shared by 6 cells: 12 x 1/6 = 22 top/bottom plane center atoms shared by 2 cells: 2 x 1/2 = 1¾ Atomic packing factor, APF = 0.74 (same as in FCC)¾ All atoms are equivalent Hexagonal Close-Packed Crystal Structure (II)ac15MSE 2090: Introduction to Materials Science Chapter 3, Structure of solids¾ Both FCC and HCP crystal structures have atomic packing factors of 0.74 (maximum possible value)¾


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