Determine the point groups of the following unit cells: a....

Determine the point groups of the following unit cells:
a. Face-centered cubic
b. Body-centered tetragonal
c. $\mathrm{CsCl} \text { (Figure } 7.7)$
d. Diamond (Figure $7.6)$
e. Nickel arsenide (Figure 7.10 )

Key Concepts

Crystallographic Point Groups

Crystallographic point groups are classifications of objects based on their symmetry elements (rotations, reflections, inversions, and improper rotations) that leave at least one point fixed. In crystallography, these groups are crucial because they describe the external symmetry of a crystal, which in turn influences many of its physical properties. Determining a crystal’s point group helps to categorize its overall symmetry and understand its behavior under different conditions.

Symmetry Operations

Symmetry operations refer to movements such as rotations, reflections, inversions, or combinations thereof that transform a structure into an indistinguishable configuration. These operations are the fundamental building blocks in the analysis of crystal symmetry. By applying these operations to a unit cell, one can classify the cell’s symmetry into a specific point group, thereby outlining the invariance properties of the crystal.

Unit Cells and Lattice Centering

A unit cell is the smallest repeatable section of a crystal lattice that captures the full symmetry and arrangement of particles in the structure. Lattice centering (such as face-centered, body-centered, or primitive) indicates additional lattice points within the cell, which significantly affect the overall symmetry. These aspects must be considered when determining a crystal’s point group, as they govern the spatial distribution of symmetry elements.

Crystal Systems

Crystal systems are broad categories into which crystals are sorted based on their unit cell dimensions and angles, such as cubic, tetragonal, hexagonal, and others. Each crystal system is associated with a distinct set of symmetry constraints and possible point groups. Understanding the crystal system of a material aids in narrowing down the potential point groups by providing initial insights into the symmetry of the unit cell.

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