Crystal Systems and Bravais Lattice Quiz

Face-Centered Cubic (FCC) Structure

• All atoms are identical in an FCC structure.
• The coordination number for FCC is 12, and there are 4 atoms per unit cell.
• The atomic packing factor (APF) for an FCC structure is 0.74, which is the maximum achievable APF.
• The close-packed directions in an FCC structure have a length of 4R, where R is the atomic radius.

Hexagonal Close-Packed (HCP) Crystal Structure

• The HCP lattice is not a Bravais lattice because not all lattice points have the same environment.
• The HCP structure has an ABAB... stacking sequence.
• The 3D projection of the HCP structure can be represented as a 2D projection with a, c, and A, B, and C sites.

Miller Indices

• Miller indices are reciprocals of the axial intercepts for a plane, cleared of fractions and common multiples.
• The algorithm to calculate Miller indices involves:
  1. Reading off intercepts of the plane with axes in terms of a, b, and c.
  2. Taking reciprocals of the intercepts.
  3. Reducing to smallest integer values.
  4. Enclosing in parentheses, with no commas.
• Examples of Miller indices include (100), (110), (111), (634), and (1011).

Crystal Systems and Bravais Lattice

• There are 7 crystal systems, which are necessary to create all point lattices.
• The 7 crystal systems are:
  1. Cubic
  2. Tetragonal
  3. Orthorhombic
  4. Rhombohedral
  5. Hexagonal
  6. Monoclinic
  7. Triclinic
• According to Bravais, there are 14 standard unit cells that can describe all possible lattice networks.
• The 14 unit cells include:
  1. Simple Cubic (SC)
  2. Body-Centered Cubic (BCC)
  3. Face-Centered Cubic (FCC)
  4. Simple Tetragonal
  5. Body-Centered Tetragonal
  6. Simple Orthorhombic
  7. Body-Centered Orthorhombic
  8. Face-Centered Orthorhombic
  9. Base-Centered Orthorhombic
  10. Simple Rhombohedral
  11. Simple Hexagonal
  12. Simple Monoclinic
  13. Base-Centered Monoclinic
  14. Simple Triclinic

Directions in Cubic Unit Cells

• Direction indices are position coordinates of the unit cell where the direction vector emerges from the cell surface, converted to integers.
• In cubic crystals, direction indices are vector components of directions resolved along each axis, reduced to the smallest integers.
• Examples of direction indices include (1,0,0) and (1,1/2,0).

HCP Crystallographic Directions

• The direction indices for HCP crystal structures can be calculated using the same method as for cubic crystals.
• Examples of HCP direction indices include [1120].

Crystallographic Planes (HCP)

• The Miller-Bravais indices for HCP crystal structures can be calculated using the same method as for cubic crystals.
• Examples of HCP Miller-Bravais indices include (1011).

Structural Difference between HCP and FCC

• The main difference between HCP and FCC structures is the arrangement of atoms in the crystal lattice.
• In HCP, the third layer of atoms is placed in the 'b' voids of the previous layer, resulting in a different stacking sequence compared to FCC.