Assume there exists some hypothetical metal that exhibits ferromagnetic behavior and that has (1) a simple cubic crystal structure (Figure 3.3), (2) an atomic radius of \(0.125 \mathrm{~nm}\), and (3) a saturation flux density of \(0.85\) tesla. Determine the number of Bohr magnetons per atom for this material

Ferromagnetism is a magnetic phenomenon exhibited by certain materials that have atomic structures allowing for collective magnetic behavior. In ferromagnetic materials, the individual magnetic moments, or spins, of atoms align in a common direction within a domain, giving rise to a strong magnetic field. This alignment is due to the exchange interaction, which is a quantum mechanical effect. The strength of a ferromagnetic material's magnetization is characterized by the saturation flux density. It indicates the maximum amount of magnetic flux that the material can output per unit area when all magnetic domains are aligned.

Understanding this behavior is essential in the context of the given hypothetical metal, as it directly relates to the calculation of the magnetic moment per atom, which tells us how many Bohr magnetons an atom contributes to the overall magnetic moment. Unlike diamagnetic and paramagnetic materials, where the response to a magnetic field is weak and tends to align in random directions, the response in ferromagnetic materials is strong and ordered.

A Bohr magneton (\textbf{\(\mu_B\)}) is the unit that measures the magnetic moment of an electron caused by its angular momentum. It is a physical constant with a value of approximately \(9.27 \times 10^{-24} \text{Am}^2\). The concept of Bohr magnetons per atom describes how many of these units make up the total magnetic moment for a single atom in the material.

In our hypothetical metal, we calculate the total magnetic moment for the entire unit cell first, then use the number of atoms in that cell to determine the magnetic moment per atom. Since ferromagnetic materials can have a high number of Bohr magnetons per atom, this contributes to their ability to be magnetized. In the exercise, by determining this number for our metal, we understand its ferromagnetic capabilities on an atomic level. Understanding how to manipulate and measure these magnetons has practical applications, such as in the design of magnetic storage devices, motors, and magnetic resonance imaging (MRI) systems.

Crystal structures refer to the highly ordered arrangement of atoms within a solid. A cubic crystal structure is one of the simplest forms of crystal lattices in which the unit cell, the smallest repetitive volume which contains the complete lattice pattern of a crystal, is shaped like a cube. There are three main types of cubic crystal structures: simple cubic, body-centered cubic, and face-centered cubic. Each has a different arrangement of atoms.

For our simple cubic structure, each corner of the cube represents the position of an atom, and thus, there is only one atom per unit cell, since each atom at a corner is shared by eight adjacent unit cells. The calculation of the volume of such a cell, based on the atomic radius, is essential in determining the magnetic moment as it allows us to quantify how much of the magnetic field is generated by the material's structure at the atomic level. This information is crucial for applications in material science, where the magnetic properties need to be designed according to the crystal structure, especially in fields dealing with magnetism and electronics.