Atomic Structures in Solid Materials

• Focus on how atoms assemble into solid structures (primarily metals), how a material's density is related to its structure, and when material properties change with sample orientation

Energy and Packing

• Dense, ordered packed structures have lower energies and increased strength
• Noncrystalline materials lack periodic packing and occur in complex structures or with rapid cooling
• "Amorphous" is synonymous with noncrystalline
• Crystalline materials feature atoms packed in periodic, 3D arrays and are typical of metals, many ceramics, and some polymers

2D and 3D Crystals

• A 2D crystal has entities arranged on a grid; the smallest repetitive unit defines its structure
• A 3D crystal extends this arrangement in three dimensions
• A crystal is an entity convolved with a grid. "Convolution" mathematically expressing how the entity is placed at every grid point to form the crystal

Crystal Systems and Metallic Structures

• The unit cell is the smallest repetitive volume containing the crystal's complete lattice pattern
• There are 7 crystal systems and 14 crystal lattices
• Lattice constants are defined by a, b, and c
• Metallic crystal structures aim to efficiently stack metal atoms
• One considers how to optimize the use of space in the stacking arrangement

Metallic Crystal Structures

• It is best to start with metallic crystals
• Covalent bonds exhibit directionality, and restricts atomic arrangement
• Similarly, ionic bonds between cations and anions limit atomic arrangement
• Metallic materials contain same nucleuses, which require to be arranged

Dense Packing in Metallic Structures

• Metallic crystal structures tend to be densely packed because:
  • Usually only one element is present, thus atomic radii are uniform
  • Metallic bonding has no direction
  • Neighboring atoms are close enough to lower bond energy
  • Electron clouds shield cores from each other
• Metals have the simplest crystal structures, and the following are typical:
  • Simple Cubic (SC)
  • Body-Centered Cubic (BCC)
  • Face-Centered Cubic (FCC)

Crystal Structure Terminology

• Face: A flat surface of a crystal Center: The central point within a crystal structure
• Edge: The line where two faces of a crystal meet Corner: The point where edges meet

Simple Cubic Structure (SC)

• SC structures are rare due to their low packing density; only Polonium (Po) adopts this structure
• Close-packed directions are along the cube edges
• Coordination number is 6, the number of nearest neighbors

Atomic Packing Factor (APF)

• APF is a ratio of the volume of atoms in a unit cell to the total volume of that cell, assuming hard spheres
• The APF for simple cubic structure = 0.52
• APF = (Volume of atoms in unit cell) / (Volume of unit cell)

Stacking for Simple Cubic Structures

• Simple cubic stacking involves stacking layers of atoms directly on top of each other

Body-Centered Cubic Structure (BCC)

• Atoms touch each other along cube diagonals
• All atoms are identical, with the center atom often shaded for clarity
• Examples include Chromium (Cr), Tungsten (W), Iron (Fe), Tantalum, and Molybdenum
• Coordination number is 8
• There are 2 atoms per unit cell: 1 at the center and 8 corners, each contributing 1/8

APF for BCC Structures

• APF for a BCC structure is 0.68
• Close-packed direction: length = 4R = √3 a, where R is the atomic radius and a is the lattice parameter
• APF = 2 * (4/3) * π * (√3a/4)³ / a³

Stacking for Body-Centered Cubic Structures

• Body-centered cubic stacking involves a more complex arrangement of atoms in layers

Face-Centered Cubic Structure (FCC)

• Atoms touch each other along face diagonals
• All atoms are identical
• Examples include Aluminum (Al), Copper (Cu), Gold (Au), Lead (Pb), Nickel (Ni), Platinum (Pt), and Silver (Ag)
• Coordination number is 12
• There are 4 atoms/unit cell: 6 face atoms x 1/2 + 8 corner atoms x 1/8

APF for FCC Structures

• APF for a face-centered cubic structure is 0.74, which is the maximum achievable APF
• Close-packed directions: length = 4R = √2 a
• A unit cell contains 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell
• APF = 4 * (4/3) * π * (√2a/4)³ / a³

FCC Stacking Sequence

• FCC stacking follows an ABCABC... pattern in its sequence

Hexagonal Close-Packed Structure (HCP)

• HCP has an ABAB... stacking sequence

Characteristics of HCP Structures

• Includes both 3D and 2D Projections
• The 2D projection features a Top, Middle, and Bottom layer.
• Coordination number is 12
• APF is 0.74
• Ratio of c/a is 1.633
• Examples include Cadmium (Cd), Magnesium (Mg), Titanium (Ti), and Zinc (Zn)

Location of Elements by Crystal Structure

• The periodic table highlights elements with face-centered cubic (fcc), body-centered cubic (bcc), and hexagonal close-packed (hcp) structures

Theoretical Density

• Density = (Number of Atoms in a unit cell * Mass of one atom) / (Total Volume of Unit Cell)
• n = number of atoms/unit cell
• Aw = atomic weight
• Vc = Volume of unit cell = a³ for cubic
• NA = Avogadro's number = 6.023 x 10²³ atoms/mol

Example Calculation: Chromium (BCC)

• A = atomic weight is 52.00 g/mol
• R = atomic radius is 0.125 nm
• n = 2 atoms/unit cell (for BCC)
• a = lattice parameter is 4R/√3 = 0.2887 nm

Density Trends

• Density order: metals > ceramics > polymers
• Metals: close-packed due to metallic bonding and have heavy atoms
• Ceramics: less dense packing and consist of lighter elements
• Polymers: have low packing density (often amorphous) and are made of light elements like C, H, and O
• Composites: intermediate values

Crystals in Engineering

• Some engineering applications require single crystals
• Examples diamond single crystals for abrasives and single-turbine blades
• Properties of crystalline materials are often related to crystal structure.
  • Quartz fractures more easily along some crystal planes than others

Polycrystalline Materials

• A diagram shows the development of polycrystalline structures, starting from individual crystals and ending with complex grain boundaries

Grain Boundaries and Sizes

• General grain sizes in engineering materials are from 1nm - 2cm.

Polycrystals

• Most engineering materials are polycrystals that are also anisotropic
• Each "grain" within a polycrystal is a single crystal
• If grains are randomly oriented, the overall component properties are not directional
• Grain sizes typically range from 1 nm to 2 cm (from a few to millions of atomic layers)

Single vs. Polycrystals

• Single Crystals have properties that vary with direction, i.e. they are anisotropic
  • Example: modulus of elasticity (E) in BCC iron is 273 GPa along the diagonal and 125 GPa along the edge
• Polycrystals have properties that may or may not vary with direction.
  • If grains are randomly oriented they have isotropic properties. (Epoly iron = 210 GPa)
  • If grains are textured, they have anisotropic properties

Polymorphism

• Polymorphism/allotropy happens when two or more distinct crystal structures exist for the same material
• e.g. Iron changes crystal structure with temperature.

Polymorphism in Tin

• Tin exhibits polymorphism, transforming between white tin (body-centered tetragonal) and grey tin (diamond cubic) at 18.2°C
• This transformation leads to ""tin disease"" or ""tin pest,"" causing degradation.

Indexing Crystal Structures

• It is important to communicate about a location, direction, or plane in a unit cell

Point Coordinates

• Point Coordinates are a common system to define points within the structure

Algorithm for Crystallographic Directions

• Vector repositioning ensures the vector passes through the origin
• Projections are read off in terms of unit cell dimensions a, b, and c
• Values are adjusted to the smallest integer
• Square brackets enclose values with no commas: [uvw]
• Negative indices are indicated with an overbar: [111]
• Families of directions are enclosed in angle brackets:

Vector Repositioning

• Sometimes requited to clearly show crystal structure

Miller Indices and Planes

• Miller Indices are reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples
• Process to determine Miller indices:
  • Read off intercepts of the plane with axes in terms of a, b, c. Take reciprocals of intercepts
  • Reduce to smallest integer values
  • Enclose in parentheses without commas, e.g., (hkl)

Families of Planes

• All parallel planes have the same Miller indices

Drawing a Plane

• To draw a plane from Miller indices:
  • Find smallest Miller index
  • Divide all 3 Miller indices by the smallest index
  • Take a reciprocal
  • Draw the intercepts on the cube

Linear Density

• Linear Density of Atoms (LD) = Number of atoms / Unit length of direction vector
• Example: For Aluminum in the [110] direction with a = 0.405 nm, # atoms = 2 and length = √2a leading to LD = 3.5 nm-1

Crystallographic Planes

• By examining the atomic packing of crystallographic planes, iron (Fe) foil can be used as a catalyst
• First, draw the (100) and (111) crystallographic planes for Fe, then calculate the planar density for each plane

Planar Density Calculation (Example)

• Planar Density = 1 / a² where a= 4√3R/3

Analyzing BCC Iron

• Higher planar density corresponds to the (111)

Determining Crystal Structure with Diffraction

• To determine unit cell dimensions (crystal structure):
  • Find out inter-planar spacing for many different plains, then back-calculate the unit cell dimensions (i.e. distance between atoms)
  • Use diffraction of light
  • Employ gratings (interplanar spacing) comparable to the wavelength of diffracted radiation

Determining Crystal Structure with Diffraction

• Diffraction requires the gratings (interplanar spacing) comparable to the wavelength of diffracted radiation

X-Ray Diffraction

• Incoming X-rays diffract from crystal planes
• Measures the critical angle for computation of planar spacing:
  • d = nλ / 2sinΘc

X-Ray Diffraction Pattern

• Diffraction patterns help identify the a-iron crystal structure.

Summary of Crystalline Solids

• Atoms assemble into crystalline or amorphous structures
• FCC, BCC, and HCP are common metallic crystal structures with associated coordination numbers and atomic packing factors. The coordination number and atomic packing factor are the same for both FCC and HCP crystal structures
• A material’s density can be predicted from its atomic weight, radius, and crystal geometry (FCC, BCC, HCP).
• Crystallographic points, directions, and planes are specified using indexing schemes, and are related to atomic linear and planar densities
• Materials can be single crystals or polycrystalline
  • Material properties generally vary with single crystal orientation, they are generally non directional
• Some materials can have more than one crystal structure, this is referred to as polymorphism/allotropy.
• X-ray diffraction is used for crystal structure and interplanar spacing determinations.