Metallic Crystal Structures and Properties

Identify and describe the three primary metallic crystal structures: Face-Centered Cubic (FCC), Body-Centered Cubic (BCC), and Hexagonal Close-Packed (HCP).
Explain the terms:
- Anisotropic and Anisotropy
- Polycrystalline and Polycrystallinity
- Polymorphic and Polymorphism
- Allotropic and Allotropy

A crystal is a solid material whose constituent atoms, molecules, or ions are arranged in an ordered pattern extending in all three spatial dimensions.
Lattice Constants describe the size and shape of the unit cell within a crystal lattice.
- a, b, c: Represent the lengths of the unit cell edges (lattice vectors).
- \alpha, \beta, \gamma: Represent the interaxial angles between these lattice vectors.

Atoms arrange themselves into crystals to minimize their free energy. This is a fundamental principle of materials science, as systems naturally tend towards lower energy states.
Structures that deviate from this ideal crystalline arrangement, such as amorphous solids or heavily disordered regions, will result in higher energies in the atoms due to less efficient packing and greater internal stresses.

Structure Description: Atoms are located at each corner of the cube and at the center of all six cube faces.
FCC Metals Examples: Aluminum (Al), Copper (Cu), Gold (Au), Lead (Pb), Nickel (Ni), Platinum (Pt), Silver (Ag).
Unit Cell: Contains 4 atoms per unit cell (calculated by (8 corner atoms * 1/8 contribution) + (6 face-centered atoms * 1/2 contribution) = 1 + 3 = 4 atoms).
Atomic Packing Factor (APF): 0.74 (meaning 74\% of the unit cell volume is occupied by atoms, the highest possible packing density for identical spheres).
Relationship between Lattice Parameter (a) and Atomic Radius (R): In an FCC structure, atoms touch along the face diagonal. Thus, (a\sqrt{2}) = 4R, which simplifies to a = 2R\sqrt{2}.
General Characteristics of FCC Metals:
- Fairly soft.
- Malleable (can be easily formed without fracturing) and typically do not harden significantly as they deform (exhibit less strain hardening).
- Will deform a great deal before failure, indicating high ductility.

Structure Description: Atoms are located at each corner of the cube with a single atom in the exact center of the cube.
BCC Metals Examples: Chromium (Cr), Iron (Fe), Molybdenum (Mo), Potassium (P, likely referring to K), Sodium (Na), Tantalum (Ta), Tungsten (W), Vanadium (V).
Unit Cell: Contains 2 atoms per unit cell ((8 corner atoms * 1/8 contribution) + (1 center atom * 1 contribution) = 1 + 1 = 2 atoms).
Atomic Packing Factor (APF): 0.68 (less dense packing than FCC).
Relationship between Lattice Parameter (a) and Atomic Radius (R): In a BCC structure, atoms touch along the body diagonal. Thus, (a\sqrt{3}) = 4R, which simplifies to a = 4R/\sqrt{3}.
General Characteristics of BCC Metals:
- Generally harder than FCC metals.
- Will not deform as much before failure (less ductile than FCC).
- Tend to get harder as they deform (exhibit more significant strain hardening).

Structure Description: A hexagonal prism with atoms at each corner of the top and bottom faces, one atom in the center of the top and bottom faces, and three atoms in the mid-plane, nestled between the top and bottom layers.
HCP Metals Examples: Cadmium (Cd), Magnesium (Mg), Cobalt (Co), Zinc (Zn), Zirconium (Zr).
Unit Cell: Contains 6 atoms per unit cell (calculated by ((12 corner atoms * 1/6 contribution) + (2 face-centered atoms * 1/2 contribution) + (3 internal atoms * 1 contribution)) = 2 + 1 + 3 = 6 atoms).
Atomic Packing Factor (APF): 0.74 (same as FCC, indicating close-packed planes).
General Characteristics of HCP Metals:
- Tend to be brittle (less ductile).
- Generally will harden rapidly as they deform (strong strain hardening).
- Cannot usually deform a lot without failure (low ductility).

Anisotropic: Describes a material whose properties depend on the direction in which they are measured. The Greek roots mean