Presentation on theme: "How do atoms ARRANGE themselves to form solids? Unit cells"— Presentation transcript:
1 How do atoms ARRANGE themselves to form solids? Unit cells Chapter OutlineHow do atoms ARRANGE themselvesto form solids?Unit cellsCrystal structuresFace-centered cubicBody-centered cubicHexagonal close-packedClose packed crystal structuresDensityTypes of solidsSingle crystalPolycrystallineAmorphous3.7–3.10 Crystallography – Not Covered / Not Tested3.15 Diffraction – Not Covered / Not TestedLearning objectives #5, #6 - Not Covered / Not TestedDensity calculation is emphasized in the tests (when you calculate density from structure – you use your understanding of structures).We have discussed atomic structure, various types of the bonds between atoms, and the correlation between the two.Now we are going to discuss the next level of structure of solid materials. The properties of solid materials not only depends on the bonding forces between atoms, but also depends on the internal structure of the solids and the arrangements of atoms in solid space.The internal structure of solids determines to a great extent the properties of solids or how a particular solid material will perform or behave in a given application.
2 Types of Solids Crystalline Amorphous SiO2 Crystalline material: periodic arraySingle crystal:periodic array over the entire extent of the materialPolycrystalline material: many small crystals or grainsAmorphous: lacks a systematic atomic arrangementCrystallineAmorphousSiO2
3 Crystal structure Consider atoms as hard spheres with a radius. Shortest distance between two atoms is a diameter.Crystal described by a lattice of points at center of atoms
4 The unit cell is building block for crystal. Repetition of unit cell generates entire crystal.Ex: 2D honeycomb represented by translation of unit cellGive an example of simple cubic lattice.Explain why one atom in honeycomb lattice can not be used as a unit cell.Ex: 3D crystalline structure
5 Metallic Crystal Structures Metals usually polycrystallineamorphous metal possible by rapid coolingBonding in metals non-directional large number of nearest neighbors and dense atomic packingAtom (hard sphere) radius, R:defined by ion core radius: ~ nmMost common unit cellsFaced-centered cubic (FCC)Body-centered cubic (BCC)Hexagonal close-packed (HCP).
6 Face-Centered Cubic (FCC) Crystal Structure (I) Atoms located at corners and on centers of facesCu, Al, Ag, Au have this crystal structureTwo representations of the FCC unit cell
7 Face-Centered Cubic Crystal Structure (II) Hard spheres touch along diagonal the cube edge length, a= 2R2The coordination number, CN = number of closest neighbors = number of touching atoms, CN = 12Number of atoms per unit cell, n = 4.FCC unit cell:6 face atoms shared by two cells: 6 x 1/2 = 38 corner atoms shared by eight cells: 8 x 1/8 = 1Atomic packing factor, APF= fraction of volume occupied by hard spheres= (Sum of atomic volumes)/(Volume of cell)= 0.74 (maximum possible)
8 Atomic Packing Fraction APF= Volume of Atoms/ Volume of CellVolume of Atoms = n (4/3) R3Volume of Cell = a3Give an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)
9 = (atoms in the unit cell, n ) x (mass of an atom, M) / Density Computations = mass/volume= (atoms in the unit cell, n ) x(mass of an atom, M) /(the volume of the cell, Vc)Atoms in the unit cell, n = 4 (FCC)Mass of an atom, M = A/NAA = Atomic weight (amu or g/mol)Avogadro number NA = 1023 atoms/molThe volume of the cell, Vc = a3 (FCC)a = 2R2 (FCC)R = atomic radiusGive an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)
10 Density ComputationsGive an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)
11 Body-Centered Cubic Crystal Structure (II) Hard spheres touch along cube diagonal cube edge length, a= 4R/3The coordination number, CN = 8Number of atoms per unit cell, n = 2Center atom not shared: 1 x 1 = 18 corner atoms shared by eight cells: 8 x 1/8 = 1Atomic packing factor, APF = 0.68Corner and center atoms are equivalent
12 Hexagonal Close-Packed Crystal Structure (I) Six atoms form regular hexagon surrounding one atom in centerAnother plane is situated halfway up unit cell(c-axis) with 3 additional atoms situated at interstices of hexagonal (close-packed) planesCd, Mg, Zn, Ti have this crystal structure
13 Hexagonal Close-Packed Crystal Structure (II) Unit cell has two lattice parameters a and c.Ideal ratio c/a = 1.633The coordination number, CN = 12 (same as in FCC)Number of atoms per unit cell, n = 6.3 mid-plane atoms not shared: 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 = 1Atomic packing factor, APF = 0.74 (same as in FCC)ca
14 Density Computations Summarized Density of a crystalline material,= density of the unit cell= (atoms in the unit cell, n ) (mass of an atom, M) / (the volume of the cell, Vc)Atoms in unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP)Mass of atom, M = Atomic weight, A, in amu (or g/mol). Translate mass from amu to grams divide atomic weight in amu by Avogadro numberNA = 1023 atoms/molVolume of the cell, Vc = a3 (FCC and BCC)a = 2R2 (FCC); a = 4R/3 (BCC)where R is the atomic radiusGive an example, ask students to compute density of gold (A=196.97, R=1.44 A, FCC)Atomic weight and atomic radius of elements are in the table in textbook front cover.
15 Face-Centered Cubic Crystal Structure (III) Corner and face atoms in unit cell are equivalentFCC has APF of 0.74Maximum packing FCC is close-packed structureFCC can be represented by a stack of close-packed planes (planes with highest density of atoms)
16 Close-packed Structures (FCC and HCP) FCC and HCP: APF =0.74 (maximum possible value)FCC and HCP may be generated by the stacking of close-packed planesDifference is in the stacking sequenceHCP: ABABAB FCC: ABCABCABC…
17 HCP: Stacking Sequence ABABAB... Third plane placed directly above first plane of atoms
18 FCC: Stacking Sequence ABCABCABC... Third plane placed above “holes” of first plane not covered by second plane
19 Polymorphism and Allotropy Some materials can exist in more than one crystal structure, Called polymorphism.If material is an elemental solid: called allotropy.Ex: of allotropy is carbon:can exist as diamond, graphite, amorphous carbon.Pure, solid carbon occurs in three crystalline forms – diamond, graphite; and large, hollow fullerenes. Two kinds of fullerenes are shown here: buckminsterfullerene (buckyball) and carbon nanotube.
20 Single Crystals and Polycrystalline Materials Single crystal: periodic array over entire materialPolycrystalline material: many small crystals (grains) with varying orientations.Atomic mismatch where grains meet (grain boundaries)Grain Boundary
21 Polycrystalline Materials Atomistic model of a nanocrystalline solid
22 Polycrystalline Materials The top figure shows a zone annealing simulation of a gas turbine blade alloy. The grain size in the initial, randomly oriented, microstructure is about ten microns. The figure shows the development of a creep resistant columnar grain structure, which results when the material is pulled through a highly localized hot zone (a narrow vertical region at the middle of the figure). Simulations allow determination of process parameters (pull speed, hot zone temperature) that optimize the grain aspect ratio (length and width).Simulation of annealing of a polycrystalline grain structure
23 Anisotropy Different directions in a crystal have different packing. For instance: atoms along the edge of FCC unit cell are more separated than along the face diagonal.Causes anisotropy in crystal propertiesDeformation depends on direction of applied stressIf grain orientations are random bulk properties are isotropicSome polycrystalline materials have grains with preferred orientations (texture): material exhibits anisotropic properties
24 Non-Crystalline (Amorphous) Solids Amorphous solids: no long-range orderNearly random orientations of atoms(Random orientation of nano-crystals can beamorphous or polycrystalline)Schematic Diagram of Amorphous SiO2
25 Summary Make sure you understand language and concepts: Allotropy AmorphousAnisotropyAtomic packing factor (APF)Body-centered cubic (BCC)Coordination numberCrystal structureCrystallineFace-centered cubic (FCC)GrainGrain boundaryHexagonal close-packed (HCP)IsotropicLattice parameterNon-crystallinePolycrystallinePolymorphismSingle crystalUnit cell