Crystal Structures and Packing Flashcards

Relationship between Unit Cell Parameters and Density

• The density ($\rho$) of a substance is determined using the molar mass ($M$), the number of particles per unit cell ($n$), the unit cell edge length ($a$), and the Avogadro number ($N_A$).
• Density formula: $\rho = \frac{M \times n}{a^3 \times N_A}$.
• For a Face-Centred Cubic (FCC) cell, the total number of particles ($n$) is 4 ($1/2 \times 6 = 3$ from faces plus particles from corners).

Problem 1.1: Density Calculation for Gold

• Given Data:     * Structure: Face-centred cubic (fcc).     * Edge length ($a$): $408\,pm = 4.08 \times 10^{-8}\,cm$.     * Molar mass ($M$): $197\,g/mol$.     * Avogadro number ($N$): $6.022 \times 10^{23}\,atoms\,mol^{-1}$.     * Number of atoms ($n$): $4$.
• Result:     * $\rho = \frac{197\,g\,mol^{-1} \times 4\,atom}{(4.08 \times 10^{-8})^3\,cm^3 \times 6.022 \times 10^{23}\,atom\,mol^{-1}}$.     * $\rho = 19.27\,g/cm^3$ (or $19.27 \times 10^3\,kg/m^3$).

Packing of Particles and Coordination Number

• Particles are treated as hard spheres to maximize interparticle attraction.
• Coordination Number: The number of neighboring spheres touching any given sphere. It measures the compactness of the structure.

Close Packed Structures in One and Two Dimensions

• One Dimension: Spheres arranged in a single row touching each other.
• Two Dimensional Square Close Packing:     * Arrangement: "AAAA…" type where spheres align horizontally and vertically.     * Coordination Number: 4.
• Two Dimensional Hexagonal Close Packing:     * Arrangement: "ABAB…" type where spheres of the second row fit into the depressions of the first.     * Coordination Number: 6.     * Features: Triangular voids (pointing up and down); more efficient than square packing.

Three Dimensional Close Packing

• Simple Cubic Lattice: Generated by stacking square close-packed layers in an "AAAA…" alignment.
• Hexagonal and Face-Centred Cubic: Generated by stacking two-dimensional hexagonal close-packed layers in two distinct ways.     * Hexagonal close packed (hcp) structure.     * Face-centred cubic (fcc) structure.

Definitions

• Unit Cell: The smallest repeating structural unit of a crystalline solid.
• Crystal Lattice: A geometrical arrangement of points (layers and corners) representing the constituent particles.