Confirm that there are \(1.72\) Bohr magnetons associated with each cobalt atom, given that the saturation magnetization is \(1.45 \times 10^{6} \mathrm{~A} / \mathrm{m}\), that cobalt has an HCP crystal structure with an atomic radius of \(0.1253 \mathrm{~nm}\) and a \(c / a\) ratio of \(1.623\).

The Bohr magneton, denoted as \(\mu_B\), is a physical constant that represents the magnetic moment of an electron orbiting a nucleus, in the absence of any external magnetic field. It's an essential unit in the study of magnetism and specifically in quantum mechanics where it acts as a basic measure of the magnetic moment. The value of the Bohr magneton is \( 9.274 \times 10^{-24} \mathrm{Am}^2 \).

As seen in the exercise, the Bohr magneton is crucial for understanding the magnetic properties of materials at the atomic level. To calculate the collective magnetic moment from an assembly of atoms or ions in a material, you would multiply the number of Bohr magnetons associated with each atom by the amount of such atoms. It's this collective behavior that determines how a material will react in the presence of an external magnetic field. Understanding the Bohr magneton helps in grasping the concept of saturation magnetization, which relates to the maximum magnetic moment a material can have.

Saturation magnetization, denoted by \(M_s\), is the maximum magnetization that a magnetic material can achieve under an external magnetic field. This indicates the maximum number of magnetic moments per unit volume aligning themselves in the direction of the magnetic field. When a material is fully magnetized, all its magnetic moments are oriented in one direction, resulting in the highest value of magnetic flux density the material can support.

In the provided exercise, saturation magnetization is used alongside the volume of an atom within a structure to calculate the number of Bohr magnetons per cobalt atom. This is important, as it shows the direct link between the physical configuration of the atoms in a material and its magnetic properties. Understanding saturation magnetization is key in material sciences, as it determines how much magnetic field a material can 'handle' before it cannot become any more magnetized.

Atoms in solids arrange themselves in various structures, one of which is the hexagonal close-packed (HCP) structure. This structure is characterized by its highly efficient packing of atoms, akin to stacking layers of spheres as tightly as possible. The HCP structure has two important parameters, \(a\) and \(c\), related to the distances between atoms in the lattice.

The 'a' parameter represents the distance between the centers of two adjacent atoms in a layer, and the 'c' parameter signifies the distance between the centers of the two opposite atoms in two stacked layers. In the context of the exercise, knowing the atomic radius and the \(c/a\) ratio is crucial to computing the unit cell volume, which in turn is necessary for further magnetic calculations. The HCP structure, with its repeating pattern, makes it possible to calculate the number of atoms per unit cell and thus find the volume occupied by a single atom. This information is fundamental when correlating atomic arrangement with magnetic behavior, especially when determining magnetization properties.