Unit Cell Models and Calculations in Solid State Chemistry

Unit Cells in Solid State Chemistry

Introduction

Unit cells are the fundamental repeating units in the crystal lattice of solids. Understanding their structure and properties is essential for analyzing the arrangement and behavior of atoms in solid materials.

• Unit Cell: The smallest repeating unit in a crystal lattice that shows the entire structure's symmetry.

• Face-Centered Cubic (FCC): A type of unit cell where atoms are located at each corner and the centers of all the faces of the cube.

• Primitive Cell: The simplest unit cell containing only one lattice point.

Pre-Lab Questions

• FCC Atoms in Unit Cell: FCC unit cell contains 4 atoms: 8 corner atoms (each shared by 8 unit cells) and 6 face atoms (each shared by 2 unit cells).

• Occupied Volume: Fraction of the unit cell's total volume that is actually filled by atoms, not just empty space.

• Coordination Number: The number of nearest neighbor atoms directly surrounding a given atom. For FCC, the coordination number is 12.

• Surrounding a Given Atom: Atoms directly adjacent to a central atom in the lattice.

Key Formulas

• FCC Cell Edge Length ($a$): $a = 2\sqrt{2}r$

• Atomic Radius ($r$): $r = \frac{a}{2\sqrt{2}}$

• Volume of Unit Cell ($V_{cell}$): $V_{cell} = a^3$

• Volume Occupied by Atoms ($V_{atoms}$): $V_{atoms} = 4 \times \frac{4}{3}\pi r^3$

• Fraction Occupied: $\text{Fraction} = \frac{V_{atoms}}{V_{cell}}$

Methods: Building Unit Cell Models

To visualize and understand the arrangement of atoms in solids, different unit cell models are constructed:

1. Make primitive cell model

2. Make face-centered cell model

3. Make body-centered cell model

Results: Unit Cell Models

Model	Description	Number of Atoms
Primitive Cell	Atoms only at the corners	1 atom per cell (each corner atom shared by 8 cells)
Face-Centered Cubic (FCC)	Atoms at corners and centers of each face	4 atoms per cell (8 corners × 1/8 + 6 faces × 1/2)
Body-Centered Cubic (BCC)	Atoms at corners and one atom at the center	2 atoms per cell (8 corners × 1/8 + 1 center)

TA Notes

• Primitive cell: 8 corner atoms, each shared by 8 unit cells, so 1 atom per cell.

• Face-centered cell: 8 corner atoms (shared by 8 cells) and 6 face atoms (shared by 2 cells), totaling 4 atoms per cell.

• Body-centered cell: 8 corner atoms (shared by 8 cells) and 1 center atom, totaling 2 atoms per cell.

Calculations in Unit Cell Analysis

Determining Cell Parameters

Calculations are essential for quantifying the properties of unit cells, such as volume, density, and atomic packing.

• Cell Edge Length ($a$): For FCC, $a = 2\sqrt{2}r$

• Volume of Unit Cell ($V_{cell}$): $V_{cell} = a^3$

• Volume Occupied by Atoms ($V_{atoms}$): $V_{atoms} = 4 \times \frac{4}{3}\pi r^3$

• Fraction Occupied: $\text{Fraction} = \frac{V_{atoms}}{V_{cell}}$

• Density ($\rho$): $\rho = \frac{\text{mass}}{\text{volume}}$

Example Calculation

• Given atomic radius $r$, calculate $a$ using $a = 2\sqrt{2}r$.

• Calculate $V_{cell}$: $V_{cell} = a^3$.

• Calculate $V_{atoms}$: $V_{atoms} = 4 \times \frac{4}{3}\pi r^3$.

• Find fraction occupied: $\text{Fraction} = \frac{V_{atoms}}{V_{cell}}$.

• Calculate density if mass and volume are known: $\rho = \frac{\text{mass}}{\text{volume}}$.

Coordination Number

• Definition: The number of nearest neighbor atoms surrounding a central atom in a crystal lattice.

• For FCC: Coordination number is 12.

• For BCC: Coordination number is 8.

References

• Alkire, L., Breen, N., and Nguyen, V., Experimental Chemistry II, 2016 (Laboratory Manual, Hayden McNeil, 2016)

• Internet sources and personal communication (as cited in original notes)

Additional info: These notes are relevant for introductory solid state chemistry and materials science, not Anatomy & Physiology. The content focuses on crystal structures, unit cell models, and related calculations, which are foundational in chemistry and physics.