Lattice Structures & Unit Cells - 16.02.26

Geometry of Spheres and Lines in Space

  • Entire radius of the forcep of spheres

  • Entire diameter of the space sphere

  • Entire radius of the sphere

  • Importance: 4 times the radius is a significant value related to geometry.

  • Definition: The type of line described is called Hypomnius.

Triangle Relationships and Algebraic Constructions

  • Important relationships exist between the sides of a triangle and algebraic formulations.

  • Notation in equations indicates values are greater than or equal to zero.

  • Given a formula relating two sides:

    • If two sides are squared, when added together and divided by 16 results in calculations leading to larger squares.

  • Calculated example yields the value 197.60 which might relate to density or other mathematical properties.

Summary of Fundamental Properties of Unit Cells

  • Summary table presented as a revision tool for unit cell configurations and their properties.

Types of Cubic Unit Cells

  1. Simple Cubic (SC)

    • Contains 1 atom per unit cell.

    • Coordination number: 6 neighbors in all directions.

  2. Body Centered Cubic (BCC)

    • Contains 2 atoms per unit cell.

    • Coordination number: 8 neighbors (1 center atom surrounded by 4 above and 4 below).

  3. Face Centered Cubic (FCC)

    • Contains 4 atoms per unit cell.

    • Coordination number: 12 neighbors.

Relationships in Radii and Distances

  • In a cubic structure:

    • Relationship generally described as 2r (radius) over the unit cell length.

    • In BCC: relationship explained as ** rac{3l}{4}** (where l is the cube edge), leading to defining distances through complex geometrical arrangements.

Diagonal Understanding in BCC

  • Calculations for body-centered cubic properties require finding a line through the center of spheres.

  • Important for understanding geometric principles that determine packing and arrangement of particles.

  • Analyzed geometrical patterns such as slicing through the diagonal of a cube which yields a length of ** ext{ extsqrt{2}}l**.

  • Result for radius calculation in spheres can be processed to yield specific relationships and densities.

Overview of Crystalline Forms and Bravais Systems

  • Total of 7 different Bravais lattices accounted, with 14 types of unit cells.

  • Properties of unit cells are shared among different systems (e.g. equivalent lengths, cuboidal shapes).

Specific Systems

  • Trigonal System: Can either have a simple or body center arrangements.

    • Unique: shares properties of cubic systems with a different edge length structure.

Historical Example - Tin Crystalline Forms

  • Tin exists in two forms: Gray tin and White tin.

  • Notable historical note refers to Napoleon’s buttons made from tin, which froze and reverted from gray to white tin below 30 degrees Celsius.

Understanding Packing Efficiency in Crystal Structures

Packing and Sphericity

  • Examination of packing efficiencies within different cubic structures.

Simple Cubic (SC)

  • Known for poor packing efficiency, leaving significant voids, around 50% of the space remains unfilled.

Body Centered Cubic (BCC)

  • Demonstrates better packing efficiency, around 68% filled.

  • Visual illustrations show that applies efficient arrangements.

Close-Packing Strategies

  • Different packing methods influence density and material organizations.

  • Denser structures can be achieved by offsetting layers, such as in:

    • Two-dimensional packing diagrams

    • Discussion of optimal arrangements using checkerboard offsets.

  • Results in close pack arrangements yield cubic structures indicated by specific layer sequences (e.g., ABC exchange sequences).

Alloy Structures and Lattice Organization

  • Two types of holes are primarily noted in crystal structures:

    • Octahedral holes: Observed within specific cubic configurations.

    • Tetrahedral holes: Found within configurations of face-centered cubic structures.

  • Filling these holes leads to unique arrangements and efficiencies based on particle sizes.

Significance of Ionic Radii and Packings

  • Relationships can significantly determine how ions arrange themselves within different compounds (e.g., Cesium Chloride CsCl).

  • Example of ionic solids discusses potential packing arrangements, the nature of interactions among the ions, and implications for stability in solid structures depending on the relative sizes and charges of ions involved (Cations vs. Anions).