Magnesium (Mg) has an HCP crystal structure and a density of \(1.74 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is the volume of its unit cell in cubic centimeters? (b) If the \(c / a\) ratio is \(1.624\), compute the values of \(c\) and \(a\).

The volume of a unit cell in a hexagonal close-packed (HCP) crystal structure is crucial because it helps us understand the arrangement of atoms in the crystal and is critical for calculating properties such as density.

In HCP structures, each unit cell's shape is determined by two lattice constants, denoted by 'a' and 'c'. 'a' represents the length of the cell's sides on the basal plane, while 'c' is the distance between the bases of the prism-shaped cell. To find the volume of such a cell, we use the established formula: \[V = \frac{3\sqrt{2}}{4}a^2c\]
These lattice constants (a and c) are not arbitrary but are connected to the atomic radius and have a fixed ratio in a perfect HCP lattice. The formula arises from the geometry of the solid and the close-packed configuration of atoms.

Density is a fundamental characteristic of materials, defined as mass per unit volume. In the context of crystal structures, it helps in identifying and comparing different materials.

To calculate the density of a crystalline substance, we make use of its molar mass and Avogadro's number to find the mass of a fixed number of atoms within the unit cell, and divide it by the unit cell volume. Applied to magnesium's HCP structure, the formula looks like this: \[\rho = \frac{m_{Mg}N}{V}\]
where \( \rho \) represents the density, \( m_{Mg} \) denotes the molar mass of magnesium, and \( N \) is the number of atoms per unit cell. For HCP, \( N \) is typically 6 as it is doubly-layered. This calculation is central to determining other structural properties and in engineering applications that depend on material strength and weight.

In crystalline structures, the lattice constants are parameters that define the unit cell's dimensions. Specifically, in a hexagonal close-packed structure, 'a' and 'c' represent the basal edge length and the height of the cell, respectively.

Understanding these lattice constants is essential as they relate to the physical dimensions of the crystal and are used to derive other properties, such as thermal expansion, slip systems for deformation, and directional dependencies in electrical or thermal conductivity. In the case of magnesium, with a given \(c/a\) ratio, combining this ratio with the calculated volume from earlier allows for the individual determination of 'a' and 'c', tying together the theoretical model with real, measurable quantities. The constants give us vital clues about the bonding strength and anisotropy in the mechanical behavior of the material.