Sodium metal crystallizes in a body centred cubic lattice with the cell edge \(a=4.29 \AA\). What is the radius of sodium atom?

In the realm of material science and solid-state physics, understanding lattice structures in solids is indispensable. These lattice structures are essentially three-dimensional arrangements of atoms, ions, or molecules in a crystal. The most common lattice structures include simple cubic, face-centered cubic, and body-centered cubic (BCC), each with its unique geometrical arrangement and coordination number, which refers to the number of nearest neighbors an atom has.

Body-centered cubic lattice, in particular, is characterized by an atom at each of the eight corners of a cube and one atom at the very center of the cube. This configuration has an important implication for the material's properties, including its density and how it interacts with light or electricity. BCC structures are prevalent in metals like iron at certain temperatures, and as we explore further, in elements like sodium. Such knowledge not only helps in identifying materials with desired properties but also in manipulating these properties for various applications.

The study of crystal structures and their properties falls under crystallography in physical chemistry. This branch of science is crucial as it allows scientists to visualize and understand the intricate arrangements of atoms within a solid. The discipline involves techniques like X-ray diffraction, which can be used to infer distances between layers of atoms and even measure the size of atoms or ions in a lattice.

This information is critical; for example, in pharmaceuticals, understanding the crystal structure of a drug can influence its efficacy and solubility. In materials science, the strength, reactivity, and other physical characteristics are often related to the arrangement of atoms in a lattice. By studying these arrangements in depth, chemists and material scientists can predict and explain the behaviors of materials under various conditions, leading to innovations in technology and industry.

Calculating the radius of an atom within a solid matrix is a fundamental task in crystallography and materials science. The radius can be inferred from the lattice parameters, which define the dimensions of the unit cell—the smallest repeating unit of the lattice. In a body-centered cubic lattice, the atom's radius is intertwined with the cube's edge length. Given the cube edge, 'a', and the lattice structure, you can employ geometric relationships to determine the atomic radius.

For BCC lattices, the body diagonal—that is, a line drawn from one corner of the cube to the opposite corner passing through the center—will be four times the atomic radius in length. The formula derived from this relationship is essential as it provides insights into the space occupied by atoms in a solid. Understanding these dimensions allows for the prediction of various physical properties, such as density and thermal expansion, all of which are crucial for material selection in engineering applications.