The empty space in this HCP unit cell is (A) \(74 \%\) (B) \(47.6 \%\) (C) \(32 \%\) (D) \(26 \%\)

Imagine lining up spheres, such as atoms, in the tightest possible arrangement; this is what we refer to as 'packing'. Now, when we talk about packing efficiency, we're measuring how much space these spheres occupy within a unit cell. Imagine a box filled with tennis balls - the more tennis balls we can fit inside without any gaps, the higher the packing efficiency will be.

In the context of crystal structures, like those found in metals and salts, a high packing efficiency means the atoms are closely packed to one another, leaving minimal empty space. This efficiency is calculated as the ratio of the combined volume of the spheres to the total volume of the unit cell, typically expressed as a percentage. In a hexagonal close-packed (HCP) structure, atoms are stacked in such a way that each atom is surrounded by six others, forming a hexagonal arrangement. This gives it a packing efficiency of around 74%, which is among the highest efficiency levels for any crystal structure.

Understanding this concept is crucial because the packing efficiency has a direct impact on the properties of a material, including its density and stability. Materials with high packing efficiencies, like certain metals, tend to be denser and less compressible.

Delving into the hexagonal close-packed structure, it's essential to visualize each layer of hexagonal-packed atoms like a tightly packed honeycomb. Each cell in the honeycomb is surrounded by neighbors, minimizing empty space. This HCP structure is one of the ways to pack spheres in the most space-efficient manner.

An HCP unit cell is a hexagonal prism composed of two hexagonal bases and six rectangular faces. In three dimensions, atoms are arranged so each is at a vertex of a regular tetrahedron, with one atom in the center of the tetrahedron. This structure is prevalent in many metals, such as magnesium and titanium, due to its high packing efficiency and the stability it confers to the atomic arrangement.

The concept of atomic packing is also critical for understanding a material’s strength and how it deforms under stress. In HCP materials, the directional bonds in this arrangement offer resistance to deformation, which is why many HCP metals are known for their hardness and durability.

When it comes to volume calculation for a hexagonal close-packed unit cell, it's necessary to use geometry and algebra. The unit cell of an HCP structure can be considered as two parts: a hexagonal prism, and a half of a hexagonal prism that captures the atoms in the staggered layers.

The volume of the hexagonal prism can be found with the formula: Volume = Area of Base × Height. The area of the hexagonal base is determined by splitting the hexagon into equilateral triangles and calculating their combined area; for an equilateral triangle, the area is \( \frac{\sqrt{3}}{4} \times a^2 \), where \( a \) is the side length. Since there are six triangles in a hexagon, we scale this area by six. The height corresponds to the distance between two consecutive hexagonal layers.

To account for the entire mass of the atoms and the full unit cell, one must consider the placement and radius of the atoms to ascertain occupied volume. Remembering that the actual volume occupied by atoms is determined by their spherical shape, you apply the sphere's volume formula, which is \( \frac{4}{3}\pi r^3 \) for each atom, where \( r \) is the atomic radius. By adding up the respective volumes of atoms fitting in the unit cell, you can calculate the total volume occupied by atoms.

Converting to Packing Efficiency

Once you have the total occupied volume, divide it by the total volume of the HCP unit cell and multiply by 100 to get the packing efficiency percentage. This allows you to understand and contextualize how much space in a material is 'filled' versus 'empty' - a critical factor for many of its physical properties.