Question

Using the projection of the unit cell for hexagonal close packing, draw four cells in a \(2 \times 2\) formation. Hence, draw the outline of the hexagon which gives hexagonal close packing its name.

Short answer

To create a hexagonal close packing (hcp) projection with a \(2\times2\) formation, first draw a hexagonal unit cell with spheres at each corner and one in the center. Create a \(2\times2\) formation by adding three adjacent hexagons. Lastly, draw the outline of the larger hexagon formed by connecting the centers of the six surrounding spheres in a layer. This hexagon encompasses the entire \(2\times2\) formation and represents the hexagonal close packing structure.

Step-by-step solution

Understanding the hcp unit cell

In hexagonal close packing, the unit cell has a hexagonal shape with one sphere at each corner and one sphere in the center of the cell. There are two additional spheres above and below the central sphere, which are shared with the neighboring cells. When looking at the projection of the unit cell, we only need to consider the contribution from the same layer where each sphere has six neighbors.

Drawing the hcp unit cell

Draw a hexagon with one sphere at each corner, and one in the center. Each sphere's center will be vertices of a smaller hexagon, which is the projection of the unit cell we are interested in. Make sure the hexagon is regular, meaning all its sides and angles are equal.

Create the \(2 \times 2\) formation

Now that we have drawn one unit cell, we need to place three more unit cells next to it in a \(2 \times 2\) formation. You can do this by drawing an adjacent hexagon to each side of the initial hexagon, sharing one side between them, and then connecting the free corners of these hexagons to create the fourth cell. You should now have a \(2 \times 2\) formation of hcp unit cells.

Draw the outline of the hexagon for hexagonal close packing

The hexagon that gives the hexagonal close packing its name is formed by connecting the centers of the six spheres surrounding a central sphere in a layer. Draw the outline of this hexagon by connecting the centers of these spheres, making sure that it is a regular hexagon with all sides and angles equal. This hexagon will be bigger than the initial hexagons you drew in step 2 and will encompass the entire \(2\times2\) formation. By following these steps, you should now have a clear understanding of the hexagonal close packing structure and a \(2\times2\) formation of hcp unit cells with the hexagon outline that gives it its name.

Key concepts

Unit Cell

In solid state chemistry, a fundamental concept is the 'unit cell', which is the smallest repeating structure in a crystal. Complex crystals are built from these unit cells, much like a large structure can be assembled from small Lego blocks. In the context of hexagonal close packing (hcp), the unit cell is a hexagonal prism comprised of layers of atoms. Each corner of the hexagon represents the position of an atom, with additional atoms at the top and bottom faces.

When constructing a model of a hcp unit cell, every edge and angle must be precisely drawn to represent the underlying symmetry of the crystal. Moreover, because hexagonal close-packed structures are a type of crystalline ordering, it's crucial to understand that the unit cell is an abstraction. In practice, you would have a vast number of these unit cells repeated in three dimensions to form a macroscopic crystal.

Visualizing and drawing the unit cell correctly is an excellent first step in comprehending more complex crystal structures, and it sets the foundation for digging deeper into the properties and behaviors of materials on a microscopic level.

Crystal Structure

Understanding crystal structure is vital within the field of solid state chemistry. Crystals are defined by their orderly and repeating pattern of atoms, ions, or molecules spread out over three dimensions. Hexagonal close packing represents one way these particles can organize themselves efficiently in space.

The defining characteristic of a crystal structure is its symmetry and how it tessellates, or tiles without gaps or overlaps. With hcp, the key feature is the hexagonal lattice system that this structure creates. Each atom is surrounded by six other atoms in the same plane, forming a regular hexagon - a foundational polygon in hexagonal close packing.

In the idealized two-dimensional projection, this creates a tiling pattern resembling a honeycomb, a real-world example of hcp. This illustrates how crystal symmetry relates to the physical properties of a material, such as its cleavage patterns, optical properties, and even its strength and durability.

Solid State Chemistry

Solid state chemistry is the study of the synthesis, structure, and properties of solid phase materials. It plays a crucial role in the development of new materials and the improvement of existing ones. Within this science, the arrangement of atoms in a solid and their interaction with each other can determine the material's electrical, magnetic, and mechanical properties, among others.

Delving into hexagonal close packing (hcp) gives us insight into the densely packed structures of metals like magnesium and titanium, explaining properties such as high strength-to-weight ratios. hcp is a prime example that showcases how the arrangement of atoms impacts the physical properties of the solid. Understanding the concept of unit cells and crystal structures is not just a theoretical exercise; it has practical implications for how we develop and use materials in various technologies.

As students of solid state chemistry, grasping how to visualize and represent these structures, like with the hcp unit cell, helps in predicting and explaining the behavior of materials under different conditions, which is fundamental for innovations in materials science.