Question

Describe the geometries of these cubic cells: simple cubic cell, body-centered cubic cell, face-centered cubic cell. Which of these cells would give the highest density for the same type of atoms?

Short answer

The geometry for the simple cubic, body-centered cubic, and face-centered cubic cells are as described. The face-centered cubic cell gives the highest density for the same type of atoms as it contains the most atoms.

Step-by-step solution

Simple Cubic Cell

A simple cubic cell has atoms at each of its 8 corners. Each corner is shared by 8 cells so a single cell contains \( \frac{1}{8}*8 = 1 \) atom.

Body-Centered Cubic Cell

A body-centered cubic cell has atoms at the 8 corners like in a simple cubic cell as well as an additional atom in the center of the cell, so a single cell contains \( \frac{1}{8}*8 + 1 = 2 \) atoms.

Face-Centered Cubic Cell

A face-centered cubic cell has atoms at each of its 8 corners and one additional atom at the center of each of the 6 faces. So a single cell contains \( \frac{1}{8}*8 + \frac{1}{2}*6 = 1+3 = 4 \) atoms. However, note that each face atom is shared by two cells.

Comparing Densities

Density depends on the number of atoms per unit volume. Since the type of atom is the same in this case, the cell with the highest number of atoms will have the highest density. Thus, the face-centered cubic cell has the highest density as it contains the most atoms.

Key concepts

Simple Cubic Cell

The simple cubic cell is the most straightforward type of crystal lattice structure. Imagine a cube with an atom located right at each of its eight corners. This arrangement looks like a box made of balls sitting at every corner. However, not all of those corner atoms belong solely to one cube. Each corner atom is shared with seven neighboring cubes, meaning only a fraction of each is inside the original cube.
To calculate, we consider that each cube hosts just an eighth (\(\frac{1}{8}\)) of each atom on its corners. Thus, when all corners are accounted for, the simple cubic cell effectively contains 1 whole atom in total:
\[\frac{1}{8} \times 8 = 1 \text{ atom} \]

• This structure is simple but not very space-efficient. Around 52% of the volume in this arrangement is empty space.
• Examples include Polonium, which crystallizes in this form.

Understanding simple cubic cells helps us grasp the basic arrangement in lattice structures and provides a steppingstone to more complex forms.

Body-Centered Cubic Cell

The body-centered cubic cell takes a step beyond the simple cubic formation. Besides the same eight corner atoms, this cell surprises by having a solitary atom nestled right in the heart of the cube. Imagine it as a friend standing dead-center inside our corner atom box.
This additional atom in the center accounts for more mass contained within just one cube. Hence, the body-centered cubic cell holds a total of two full atoms:
\[\frac{1}{8} \times 8 + 1 = 2 \text{ atoms} \]

• Impressively, the body-centered cubic structure utilizes space more efficiently, with about 68% of its space packed.
• Many metals such as Iron at room temperature, Tungsten, and Chromium exhibit this type of crystal structure.

This cell enhances understanding of cubic structures by showing how additional atoms can be incorporated to increase density even within the same outline.

Face-Centered Cubic Cell

Finally, the face-centered cubic cell, renowned for its density, further builds on complexity. This cell encompasses an atom at each corner similar to the previous two forms, but it also features an extra atom right at the center of each of its six faces.
Each face atom faces into this cube and is shared between two adjoining cells. As a result, each face atom contributes just half (\(\frac{1}{2}\)) of itself to the cell. Altogether, this means:
\[\frac{1}{8} \times 8 + \frac{1}{2} \times 6 = 1 + 3 = 4 \text{ atoms} \]

• The face-centered structure is extremely efficient, making use of approximately 74% of the space, minimizing empty volume.
• This is why metals like Copper, Aluminum, and Gold prefer this crystal lattice.

This densest structure fascinates with its efficiency and adds an elegant complexity to the cubic cell family, capturing how adjustments in atom positioning can lead to dramatic differences in material characteristics.