# Chapter 3: Structure of Metals and Ceramics. Chapter 3: Structure of Metals and Ceramics. Learning Objective

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1 Chapter 3: Structure of Metals and Ceramics Chapter 3: Structure of Metals and Ceramics Goals Define basic terms and give examples of each: Lattice Basis Atoms (Decorations or Motifs) Crystal Structure Unit Cell Coordination Numbers Describe hard-sphere packing and identify cell symmetry. Crystals density: the mass per volume (e.g. g/cm 3 ). Linear Density: the number of atoms per unit length (e.g. cm -1 ). Planar Densities: the number of atoms per unit area (e.g. cm -2 ). Learning Objective Know and utilize definitions to describe structure and defects in various solid phases (crystal structures). Compute densities for close-packed structures. Identify Symmetry of Cells. Specify directions and planes for crystals and be able to relate to characterization experiments. ENEGY AND PACKING Atomic PACKING Non dense, random packing Dense, regular packing Dense, regular-packed structures tend to have lower energy. Crystalline materials... atoms pack in periodic, 3D arrays typical of: -metals -many ceramics -some polymers Noncrystalline materials... atoms have no periodic packing occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline crystalline SiO2 Adapted from Fig. 3.18(a), Callister 6e. noncrystalline SiO2 Adapted from Fig. 3.18(b), Callister 6e. From Callister 6e resource CD. 1

2 It s geometry! Crystalline Solids: Unit Cells Unit Cell: The basic structural unit of a crystal structure. Its geometry and atomic positions define the crystal structure. A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together with purely translational repetition. Note: More than one unit cell can be chosen for a given crystal structure but by convention/convenience the one with the highest symmetry is chosen. Fig. 3.1 Atomic configuration in Face-Centered-Cubic Arrangement a Crystalline Solids: Unit Cells A Space LATTICE is an infinite, periodic array of mathematical points, in which each point has identical surroundings to all others. LATTICE BASIS Several GIFS that follow were taken from Dr. Heyes (Oxford) excellent webpage. A CYSTAL STUCTUE is a periodic arrangement of atoms in the crystal that can be described by a LATTICE + ATOM DECOATION (called a BASIS). Crystalline Solids: Unit Cells Unit Cells and Unit Cell Vectors Important Note: Lattice points are a purely mathematical concept, whereas atoms are physical objects. So, don't mix up atoms with lattice points. Lattice Points do not necessarily lie at the center of atoms. For example, the only element exhibiting Simple Cubic structure is Po. In Figure (a) is the 3-D periodic arrangement of Po atoms, and Figure (b) is the corresponding space lattice. In this case, atoms lie at the same point as the space lattice. c v b v Lattice parameters axial lengths: a, b, c interaxial angles: α, β, γ unit vectors: a v b v c v In general: a b c α β γ a v All period unit cells may be described via these vectors and angles. 2

3 Possible Crystal Classes Possible Crystal Classes Unit Cells Types A unit cell is the smallest component of the crystal that reproduces the whole crystal when stacked together with purely translational repetition. Primitive (P) unit cells contain only a single lattice point. Internal (I) unit cell contains an atom in the body center. Face (F) unit cell contains atoms in the all faces of the planes composing the cell. Centered (C) unit cell contains atoms centered on the sides of the unit cell. Primitive Face-Centered Unit Cells Types Often it s convenient to define a non-primitive unit cell to reveal overtly the higher symmetry. Then, one has to count carefully "how many atoms are in unit cell" (see next). Face-Centered Primitive (with 1 atom/cell, no symmetry) Cube (showing cubic symmetry w/ 4atoms/cell) Body-Centered End-Centered Combining 7 Crystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monclinic, triclinic, trigonal) with 4 unit cell types (P, I, F, C) symmetry allows for only 14 types of 3-D lattice. Sometimes it is convenient to define a non-primitive unit cell to reveal overtly the higher symmetry. Then, one has to count carefully "how many atoms are in unit cell" (see next). Combining 7 Crystal Classes (cubic, tetragonal, orthorhombic, hexagonal, monclinic, triclinic, trigonal) with 4 unit cell types (P, I, F, C) symmetry allows for only 14 types of 3-D lattice. KNOW THIS! Combining these 14 Bravais lattices with all possible symmetry elements (such as rotations, translations, mirrors, glides, etc.) yields 230 different Space Groups! 3

4 The 14 Bravais Lattices! Counting Number of Atoms Per Unit Cell Simple 2D Triangular Lattice Lattice showing primitive unit cell (in red) and a square, non-primitive unit cell (in green). Self-Assessment: Why can't the blue triangle be a unit cell? Counting Lattice Points/Atoms in 2D Lattices Unit cell is Primitive (1 lattice point) but contains 2 atoms in the Basis. Atoms at the corner of the 2D unit cell contribute only 1/4 to unit cell count. Atoms at the edge of the 2D unit cell contribute only 1/2 to unit cell count. Atoms within the 2D unit cell contribute 1 as they are entirely contained inside. UNIT CELL - 3D Lattices Counting Number of Atoms Per Unit Cell Counting Atoms in 3D Cells Atoms in different positions are shared by differing numbers of unit cells. Vertex atom shared by 8 cells => 1/8 atom per cell. Edge atom shared by 4 cells => 1/4 atom per cell. Face atom shared by 2 cells => 1/2 atom per cell. Body unique to 1 cell => 1 atom per cell. Simple Cubic 8 atoms but shared by 8 unit cells. So, 8 atoms/8 cells = 1 atom/unit cell How many atoms/cell for Body-Centered Cubic? And, Face-Centered Cubic? 4

5 Coordination Number of a Given Atom Unit Cells and Volume Packing Number of nearest-neighbor atoms What are basic structural parameters, e.g. lattice constant or side of cube? How many atoms per cell? What is volume per cell? What is the atomic packing factor (APF)? What is the closed-packed direction? What are (linear) densities of less close-packed directions? What are planar densities of every plane? Atomic configuration in Face-Centered-Cubic Arrangement a It s all geometry. Need to relate cube dimension a to Packing of ideal spherical atoms of radius. Simple cubic: coordination number, CN = 6 2a = 4 Atomic Packing Fraction for FCC Basic Geometry for FCC APF = vol. of atomic spheres in unit cell total unit cell vol. Face-Centered-Cubic Arrangement Geometry along close-packed direction give relation between a and a 2a = Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell Depends on: Crystal structure. How close packed the atoms are. In simple close-packed structures with hard sphere atoms, independent of atomic radius V a 3 3 unit _ cell = a = 2 2) = 16 2 ( Geometry: 4 atoms/unit cell 3 a = 2 2 Coordination number = 12 V atoms 4 3 = 4 π 3 5

6 Atomic Packing Fraction for FCC Summary APF for BCC APF = vol. of atomic spheres in unit cell total unit cell vol. How many spheres (i.e. atoms)? 4/cell What is volume/atom? 4π 3 /3 What is cube volume/cell? a 3 How is related to a? 2a = 4 Face-Centered-Cubic Arrangement Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell = 0.74 Independent of! Again, geometry along close-packed direction give relation between a and. a 2a 4 a 3 Geometry: 2 atoms/unit cell 4 a 3 Coordination number = APF = V 3 π a atoms = V cell a 3 = 3π 8 = 0.68 FCC Stacking A B C Highlighting the stacking ABCABC... repeat along <111> direction gives Cubic Close-Packing (CCP) Face-Centered-Cubic (FCC) is the most efficient packing of hard-spheres of any lattice. Unit cell showing the full symmetry of the FCC arrangement : a = b =c, angles all 90 4 atoms in the unit cell: (0, 0, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0) Self-Assessment: Write FCC crystal as BCT unit cell. Highlighting the faces 6

7 A less close-packed structure is Body-Centered-Cubic (BCC). Besides FCC and HCP, BCC structures are widely adopted by metals. FCC Unit Cell Highlighting the ABC planes and the cube. Highlighting the hexagonal planes in each ABC layer. Unit cell showing the full cubic symmetry of the BCC arrangement. BCC: a = b = c = a and angles α = β =γ= atoms in the cubic cell: (0, 0, 0) and (1/2, 1/2, 1/2). Body-Centered-Cubic (BCC) can be template for more Complex Structures: Lattice with Basis Atoms Lattice points in space decorated with buckeyballs or viruses. ABABAB... repeat along <111> direction gives Hexagonal Close-Packing (HCP) Unit cell showing the full symmetry of the HCP arrangement is hexagonal Hexagonal: a = b, c = 1.633a and angles α = β = 90, γ = atoms in the smallest cell: (0, 0, 0) and (2/3, 1/3, 1/2). 7

8 HCP Stacking A Comparing the FCC and HCP Planes Stacking Looking down (0001) plane Highlighting the stacking B FCC HCP A Layer A Layer B Highlighting the cell Figure 3.3 Self-Assessment: How many atoms/cell? MSE 280: Introduction to Engineering Materials Looking down (111) plane! Layer A D.D. Johnson 2004, MSE 280: Introduction to Engineering Materials Linear Density in FCC Packing Densities in Crystals: Lines Planes and Volumes LD = Concepts FCC D.D. Johnson 2004, Number of atoms centered on a direction vector Length of the direction vector Linear Density: No. of atoms along a direction vector per length of direction vector Planar Density: No. of atoms per area of plane per area of plane Example: Calculate the linear density of an FCC crystal along [1 1 0]. ASK a. How many spheres along blue line? b. What is length of blue line? Versus ANSWE a. 2 atoms along [1 1 0] in the cube. b. Length = 4 Linear and Planar Packing Density which are independent of atomic radius! LD110 = 2atoms 1 = 4 2 Also, Theoretical Density XZ = 1i + 1j + 0k = [110] Self-assessment: Show that LD 100 = 2/4. MSE 280: Introduction to Engineering Materials D.D. Johnson 2004, MSE 280: Introduction to Engineering Materials D.D. Johnson 2004,

9 Linear Packing Density in FCC Planar Density in FCC LDP= Number of radii along a direction vector Length of the direction vector PD = Number of atoms centered on a given plane Area of the plane Example: Calculate the LPD of an FCC crystal along [1 1 0]. Example: Calculate the PD on (1 1 0) plane of an FCC crystal. ASK a. How many radii along blue line? b. What is length of blue line? ANSWE a. 2 atoms * 2. b. Length = 4 Count atoms within the plane: 2 atoms Find Area of Plane: Self-assessment: Show that LPD 100 = 2/2. LPD 110 = 2 * 2 4 = 1 Fully CLOSE-PACKED. Always independent of! Hence, a = PD = = Planar Packing Density in FCC Theoretical Density, ρ Area of atoms centered on a given plane PPD = Area of the plane Example: Calculate the PPD on (1 1 0) plane of an FCC crystal. 4 a = 2 2 Hence, Find area filled by atoms in plane: 2π 2 Find Area of Plane: PPD = 2π = π = Always independent of! Self-assessment: Show that PPD 100 = π/4 = Example: Copper Data from Table inside front cover of Callister (see next slide): crystal structure = FCC: 4 atoms/unit cell atomic weight = g/mol (1 amu = 1 g/mol) atomic radius = nm (1 nm = 10 cm)-7 esult: theoretical ρcu = 8.89 g/cm 3 Compare to actual: ρcu = 8.94 g/cm 3 9

10 Element Aluminum Argon Barium Beryllium Boron Bromine Cadmium Calcium Carbon Cesium Chlorine Chromium Cobalt Copper Flourine Gallium Germanium Gold Helium Hydrogen Characteristics of Selected Elements at 20 C Symbol Al Ar Ba Be B Br Cd Ca C Cs Cl Cr Co Cu F Ga Ge Au He H At. Weight (amu) Density (g/cm 3 ) Atomic radius (nm) Adapted from Table, "Characteristics of Selected Elements", inside front cover, Callister 6e. DENSITIES OF MATEIAL CLASSES ρ metals ρ ceramics ρ polymers Metals have... close-packing (metallic bonds) large atomic mass Ceramics have... less dense packing (covalent bonds) often lighter elements Polymers have... poor packing (often amorphous) lighter elements (C,H,O) Composites have... intermediate values Data from Table B1, Callister 6e. SUMMAY Materials come in Crystalline and Non-crystalline Solids, as well as Liquids/Amoprhous. Polycrystals are important. Crystal Structure can be defined by space lattice and basis atoms (lattice decorations or motifs). Only 14 Bravais Lattices are possible. We focus only on FCC, HCP, and BCC, I.e., the majority in the periodic table and help determine most CEAMIC structures. Crystal types themselves can be described by their atomic positions, planes and their atomic packing (linear, planar, and volumetric packing fraction). We now know how to determine structure mathematically. So how to we do it experimentally? DIFFACTION. 10

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