Chapter 3: The Structure of Crystalline Solids ISSUES TO ADDRESS... • How do atoms assemble into solid structures? • How does the density of a material depend on its structure? • When do material properties vary with the sample (i.e., part) orientation?

The “unit cell” is the basic repeating unit of the arrangement of atoms, ions or molecules in a crystalline solid. The “lattice” refers to the 3-D array of particles in a crystalline solid. One type of atom occupies a “lattice point” in the array.

Examples of Unit Cells

  SPACE LATTICE AND UNIT CELLS   Space Lattice atoms arranged in a pattern that repeats itself in three dimensions.   Unit cell smallest grouping which can be translated in three dimensions to recreate the space lattice.    

Crystalline Versus Amorphous Quartz Obsidian

Crystals

Crystal Structures Atoms (and later ions) will be viewed as hard spheres. In the case of pure metals, the packing pattern often provides the greatest spatial efficiency (closest packing). Ionic crystals can often be viewed as a close-packed arrangement of the larger ion, with the smaller ion placed in the “holes” of the structure.

Unit Cells Crystals consist of repeating asymmetric units which may be atoms, ions or molecules. The space lattice is the pattern formed by the points that represent these repeating structural units.

Unit Cells A unit cell of the crystal is an imaginary parallel-sided region from which the entire crystal can be built up. Usually the smallest unit cell which exhibits the greatest symmetry is chosen. If repeated (translated) in 3 dimensions, the entire crystal is recreated.

Crystal Structures Types of crystal structures Face centered cubic (FCC) Body centered cubic (BCC) Hexagonal close packed (HCP)

3 2 1 1 4 2 1

BCC FCC HCP

Face Centered Cubic (FCC) Atoms are arranged at the corners and center of each cube face of the cell. Atoms are assumed to touch along face diagonals

Face Centered Cubic (FCC) The lattice parameter, a, is related to the radius of the atom in the cell through: Coordination number: the number of nearest neighbors to any atom. For FCC systems, the coordination number is 12.

Face Centered Cubic (FCC) Atomic Packing Factor: the ratio of atomic sphere volume to unit cell volume, assuming a hard sphere model. FCC systems have an APF of 0.74, the maximum packing for a system in which all spheres have equal diameter.

Body Centered Cubic Atoms are arranged at the corners of the cube with another atom at the cube center.

ATOMIC PACKING FACTOR: BCC Adapted from Fig. 3.2, Callister 6e. • APF for a body-centered cubic structure = p3/8 = 0.68

Body Centered Cubic Since atoms are assumed to touch along the cube diagonal in BCC, the lattice parameter is related to atomic radius through:

Principal Metallic Crystal Structures   We will concentrate on three of the more densely packed crystal structures, BCC - body centered cubic, FCC - face centered cubic, and HCP - hexagonal close packed.   BCC - 2 atoms per unit cell CN = 8 

FCC - 4 atoms per unit cell CN = 12  HCP - 6 atoms per unit cell CN = 12

Hexagonal Close Packed Cell of an HCP lattice is visualized as a top and bottom plane of 7 atoms, forming a regular hexagon around a central atom. In between these planes is a half-hexagon of 3 atoms.

Close Packing Since metal atoms and ions lack directional bonding, they will often pack with greatest efficiency. In close or closest packing, each metal atom has 12 nearest neighbors. The number of nearest neighbors is called the coordination number. Six atoms surround an atom in the same plane, and the central atom is then “capped” by 3 atoms on top, and 3 atoms below it.

Close Packing

# of Atoms/Unit Cell Atoms in corners are ⅛ within the cell For atoms in a cubic unit cell: Atoms in corners are ⅛ within the cell

# of Atoms/Unit Cell Atoms on faces are ½ within the cell For atoms in a cubic unit cell: Atoms on faces are ½ within the cell

# of Atoms/Unit Cell A face-centered cubic unit cell contains a total of 4 atoms: 1 from the corners, and 3 from the faces.

# of Atoms/Unit Cell For atoms in a cubic unit cell: Atoms in corners are ⅛ within the cell Atoms on faces are ½ within the cell Atoms on edges are ¼ within the cell

Other Metallic Crystal Structures Body-centered cubic unit cells have an atom in the center of the cube as well as one in each corner. The packing efficiency is 68%, and the coordination number = 8.

Other Metallic Crystal Structures Simple cubic (or primitive cubic) unit cells are relatively rare. The atoms occupy the corners of a cube. The coordination number is 6, and the packing efficiency is only 52.4%.

Polymorphism Two or more distinct crystal structures for the same material (allotropy/polymorphism)     ex. titanium   , -Ti ex. carbon (cubic)diamond, (hexagonal)graphite BCC FCC 1538ºC 1394ºC 912ºC -Fe -Fe -Fe liquid Ex.iron system

Polymorphism Many metals exhibit different crystal structures with changes in pressure and temperature. It is temperature /pressure dependent phenomenon.

Example: Iron. liquid above 1539 C Example:   Iron liquid above 1539 C. δ-iron (BCC) between 1394 and 1539 C. γ-iron (FCC) between 912 and 1394 C. α-iron (BCC) between -273 and 912 C.    

Single crystal All unit cells interlock in the same way and have the same orientation. Single crystals exist in nature, but they may also be produced artificially. They are ordinarily difficult to grow, because the environment must be carefully controlled.

Polycrystalline Materials Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline The Various stages in the solidification of a polycrystalline specimen are: Initially, small crystals or nuclei form at various positions. The small grains grow by the successive addition from the surrounding liquid of atoms to the structure of each. The extremities of adjacent grains impinge on one another as the solidification process approaches completion exists some atomic mismatch within the region where two grains meet; this area, called a grain boundary

Crystals as Building Blocks • Some engineering applications require single crystals: --diamond single crystals for abrasives --turbine blades • Properties of crystalline materials often related to crystal structure. --Ex: Quartz fractures more easily along some crystal planes than others.

Polycrystals Anisotropic • Most engineering materials are polycrystals. 1 mm Isotropic • Each "grain" is a single crystal. • If grains are randomly oriented, overall component properties are not directional. • Grain sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

Single vs Polycrystals E (diagonal) = 273 GPa E (edge) = 125 GPa • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron: • Polycrystals 200 mm -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa) -If grains are textured, anisotropic.

Theoretical Density, r A knowledge of crystal structure of a metallic solid permits computation density : Cell Unit of Volume Total in Atoms Mass Density =  = VC NA n A  = where n = number of atoms/unit cell A = atomic weight g/mole VC = Volume of unit cell = a3 for cubic NA = Avogadro’s number = 6.023 x 1023 atoms/mol

Theoretical Density, r  = a R Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n = 2 a = 4R/ 3 = 0.2887 nm  = a 3 52.00 2 atoms unit cell mol g volume 6.023 x 1023 theoretical = 7.18 g/cm3 ractual = 7.19 g/cm3

Densities of Material Classes In general Graphite/ r metals r ceramics r polymers Metals/ Composites/ > > Ceramics/ Polymers Alloys fibers Semicond 30 Why? Metals have... • close-packing (metallic bonding) • often large atomic masses 2 Magnesium Aluminum Steels Titanium Cu,Ni Tin, Zinc Silver, Mo Tantalum Gold, W Platinum *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers 10 in an epoxy matrix). G raphite Silicon Glass - soda Concrete Si nitride Diamond Al oxide Zirconia Ceramics have... • less dense packing • often lighter elements 5 3 4 (g/cm ) 3 Wood AFRE * CFRE GFRE* Glass fibers Carbon fibers A ramid fibers r H DPE, PS PP, LDPE PC PTFE PET PVC Silicone Polymers have... • low packing density (often amorphous) • lighter elements (C,H,O) 2 1 Composites have... • intermediate values 0.5 0.4 0.3

Energy and Packing • Non dense, random packing typical neighbor bond length bond energy • Dense, ordered packing Energy r typical neighbor bond length bond energy Dense, ordered packed structures tend to 1.have lower energies and more stable energy arrangements. 2. Atoms come closer together and bond more tightly

Attractive Energy: the energy released when the ions come close together Repulsive energy: the energy absorbed as the ions come close together Net Energy: the sum of energies associated with the attraction and repulsion of the ions It is minimum when the ions are at their equilibrium separation distance r0. At the minimum energy, the force between the ions are zero. The ions will remain at an equilibrium separation distance r0 ( more stable).

Zincblende/Diamond Lattices The Cubic Unit Cell Zincblende Lattice The Cubic Unit Cell Other views of the cubic unit cell

Diamond Lattice Diamond Lattice The Cubic Unit Cell

Zincblende (ZnS) Lattice Zincblende Lattice The Cubic Unit Cell.

Wurtzite Structure Wurtzite Structure We’ve also seen: Many semiconductors have the Wurtzite Structure Tetrahedral coordination: Each atom has 4 nearest-neighbors (nn). Basis set: 2 atoms. Primitive lattice  hexagonal close packed (hcp). 2 atoms per hcp lattice point A Unit Cell looks like

Materials and Packing Si Oxygen Crystalline materials... • atoms pack in periodic, 3D arrays • typical of: -metals -many ceramics -some polymers crystalline SiO2 Si Oxygen Noncrystalline materials... • atoms have no periodic packing • occurs for: -complex structures -rapid cooling "Amorphous" = Noncrystalline noncrystalline SiO2

Crystal: is a solids in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. Long range order exist Noncrystalline(amorphous): materials that don’t crystallize, this long range atomic order is absent Crystal structure: it is the manner in which atoms, ions, or molecules are spatially arranged. Or A regular 3 dimensional pattern of atoms or ions in space There are an extremely large number of different crystal structures all having long range atomic order. Crystal system: is described in terms of the unit cell geometry. A crystal structure is described by both the geometry of, and atomic arrangements within the unit cell, whereas a Crystal system is described only in terms of the unit cell geometry. For example, face-centered cubic and body-centered cubic are crystal structures that belong to the cubic crystal system.

Crystal Systems Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems 14 crystal lattices a, b, and c are the lattice constants a, b, c, α, β, γ are lattice parameters

3 2 1 1 4 2 1

Section 3.4 – Metallic Crystal Structures How can we stack metal atoms to minimize empty space? 2-dimensions vs. Now stack these 2-D layers to make 3-D structures

Metallic Crystal Structures • Tend to be densely packed. • 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. - Electron cloud shields cores from each other • Have the simplest crystal structures. We will examine three such structures...

Simple Cubic Structure (SC) • Rare due to low packing density (only Po has this structure) • Close-packed directions are cube edges. • Coordination # = 6 (# nearest neighbors)

Atomic Packing Factor (APF) Volume of atoms in unit cell* APF = Volume of unit cell *assume hard spheres • APF for a simple cubic structure = 0.52 close-packed directions a R=0.5a contains 8 x 1/8 = 1 atom/unit cell atom volume atoms unit cell 4 3 p (0.5a) 1 APF = 3 a unit cell volume

Body Centered Cubic Structure (BCC) • Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing. ex: Fe (), Tantalum, Molybdenum • Coordination # = 8 2 atoms/unit cell: 1 center + 8 corners x 1/8

Atomic Packing Factor: BCC • APF for a body-centered cubic structure = 0.68 a R a 3 a a 2 length = 4R = Close-packed directions: 3 a Adapted from Fig. 3.2(a), Callister 7e. APF = 4 3 p ( a/4 ) 2 atoms unit cell atom volume a

Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals. --Note: All atoms are identical; the face-centered atoms are shaded differently only for ease of viewing. ex: Al, Cu, Pb, Ni, Ag • Coordination # = 12 4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

Atomic Packing Factor: FCC • APF for a face-centered cubic structure = 0.74 a 2 a maximum achievable APF Close-packed directions: length = 4R = 2 a Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell APF = 4 3 p ( 2 a/4 ) atoms unit cell atom volume a

CLOSE-PACKED CRYSTAL STRUCTURES FOR METALS Both face-centered cubic and hexagonal close-packed crystal structures have atomic packing factors of 0.74, which is the most efficient packing of equal sized spheres or atoms. In addition to unit cell representations, these two crystal structures may be described in terms of close-packed planes of atoms (i.e., planes having a maximum atom or sphere-packing density); a portion of one such plane is illustrated in Figures bellow. Both crystal structures may be generated by the stacking of these close-packed planes on top of one another; the difference between the two structures lies in the stacking sequence.

Hexagonal Close-Packed Structure (HCP) • ABAB... Stacking Sequence • 3D Projection • 2D Projection c a A sites B sites Bottom layer Middle layer Top layer • Coordination # = 12 6 atoms/unit cell • APF = 0.74 ex: Cd, Mg, Ti, Zn • c/a = 1.633

CRYSTALLOGRAPHIC POINTS, DIRECT IONS AND PLANES Crystallographic planes and directions are specified in terms of an indexing scheme. Crystallographic directional and planar equivalencies are related to atomic linear and planar densities, respectively. The atomic packing (i.e., planar density) of spheres in a crystallographic plane depends on the indices of the plane as well as the crystal structure.

Point Coordinates To specify the position of any point located within unit cell. z x y a b c 000 111 Point coordinates for unit cell center are a/2, b/2, c/2 ½ ½ ½ Point coordinates for unit cell corner are 111 Translation: integer multiple of lattice constants  identical position in another unit cell Lattice constants: a, b, c z 2c y b b

Linear Density 3.5 nm a 2 LD = Linear Density  LD = [110] j a [110] ex: linear density of Al in [110] direction  a = 0.405 nm ½+1+ ½=2 atoms # atoms length 1 3.5 nm a 2 LD - = LD is important relative to the process of slip-the mechanism by which metals plastically deform

Planar Density of (100) Iron Solution:  At T < 912C iron has the BCC structure. 2D repeat unit R 3 4 a = (100) Radius of iron R = 0.1241 nm = Planar Density = a 2 1 atoms 2D repeat unit nm2 12.1 m2 = 1.2 x 1019 R 3 4 area

Planar Density of (111) Iron Solution (cont):  (111) plane 1 atom in plane/ unit surface cell 2 a atoms in plane atoms above plane atoms below plane 2D repeat unit 3 h = a 2 3 2 R 16 4 a ah area = ÷ ø ö ç è æ 1 = nm2 atoms 7.0 m2 0.70 x 1019 3 2 R 16 Planar Density = 2D repeat unit area

SUMMARY • Atoms may assemble into crystalline or amorphous structures. • Common metallic crystal structures are FCC, BCC, and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal 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). • Crystallographic points, directions and planes are specified in terms of indexing schemes. Crystallographic directions and planes are related to atomic linear densities and planar densities.

SUMMARY • Materials can be single crystals or polycrystalline. Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains. • Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy). • X-ray diffraction is used for crystal structure and interplanar spacing determinations.

Alloys Alloys are solid solutions of metals. They are usually prepared by mixing molten components. They may be homogeneous, with a uniform distribution, or occur in a fixed ratio, as in a compound with a specific internal structure.