Iron crystallizes in a bec structure. The atomic radius of iron is \(124 \mathrm{pm}\). Determine (a) the number of atoms per unit cell; (b) the coordination number of the lattice; (c) the length of the side of the unit cell.

Crystallography is the science that examines the arrangement of atoms in crystalline solids. One common structure that emerges in the realm of crystallography is the body-centered cubic (bcc) structure, which is of particular interest when exploring the properties of metals, such as iron.

In this structure, atoms are positioned at each corner of a cube and a single atom is located at the very center, resulting in a distinctive geometric pattern. The bcc arrangement contributes significantly to the physical characteristics of a metal, such as its density and how it interacts with light. The study of crystallography not only aids in identifying these structures but also in comprehending the behaviors of materials on an atomic level, influencing how they are utilized in various applications.

The concept of the coordination number is a central pillar in understanding crystal structures. It signifies the number of nearest-neighbor atoms that surround a given atom in a crystal lattice.

For the body-centered cubic structure, the coordination number is 8. This illustrates that each atom is immediately surrounded by 8 other atoms: 4 in the same plane and an additional 4 from the adjacent planes above and below. This number is not just a count; it intimates the degree of an atom’s interaction with its surrounding neighbors, playing a critical role in determining the physical and chemical properties of a material, such as its stability, melting point, and possible reactivity.

When examining the microscopic world of crystals, 'unit cell geometry' emerges as a fundamental concept. It delineates the smallest repeating unit that mirrors the entirety of a crystal's structure, often visualized as a geometric shape like a cube in the case of a bcc structure.

The unit cell of the bcc crystal is pivotal because it defines the spatial arrangement of the atoms and the dimensions of the crystal as a whole. The side length of this cube, often referred to as 'a', can be mathematically related to the atomic radius through the cube's body diagonal, which is the line connecting opposite corners passing through the center. Understanding these geometric relationships enables the calculation of critical material constants such as density, and it provides insight into mechanical properties like the material's response to stress.