Question

Vanadium crystallizes in a body-centered cubic lattice (the \(\mathrm{V}\) atoms occupy only the lattice points). How many \(\mathrm{V}\) atoms are present in a unit cell?

Short answer

There are 2 vanadium atoms in a unit cell of a body-centered cubic lattice.

Step-by-step solution

Understanding the Structure

Vanadium forms a body-centered cubic (BCC) unit cell structure. This means that there are atoms at each of the eight corners of the cubic unit cell and a single atom at the center.

Corner Atoms Contribution

In a cubic unit cell, each atom at a corner is shared among eight neighboring unit cells. Therefore, the contribution of each corner atom to a single unit cell is \( \frac{1}{8} \). As there are eight corner atoms in a BCC structure, their total contribution is \( 8 \times \frac{1}{8} = 1 \) atom.

Center Atom Contribution

A BCC structure has one atom entirely located at the center of the unit cell. This atom is not shared with any other unit cell, so its contribution is 1 whole atom.

Calculating Total Atoms in Unit Cell

To find the total number of vanadium atoms in the unit cell, add the contributions of the corner atoms and the center atom. The total is \( 1 + 1 = 2 \) atoms.

Key concepts

Vanadium Crystallization

In the world of crystals, different elements demonstrate unique patterns in how they organize themselves at the microscopic level. Vanadium is fascinating in this regard because it forms what's known as a body-centered cubic (BCC) lattice. To visualize this, imagine a cube. Vanadium atoms are situated not only at each of the cube's corners but also occupy a central spot within the cube itself.
This arrangement is not only aesthetically intriguing but also plays a critical role in determining the material properties of vanadium, such as its strength and malleability. Crystallization is essential for materials science applications, including vanadium's use in alloys that strengthen steel.

Unit Cell Structure

The unit cell is the smallest repeating unit that defines the entire crystal structure and serves as a building block for creating the larger structure. For vanadium, this is characterized as a body-centered cubic unit cell. The BCC is one of the simplest and most common lattice structures, consisting of atoms at each of the eight corners with a single atom entirely located in the center.
Each corner atom touches neighboring cells as well, making the actual unit cell more interconnected than it might initially seem. Understanding the specifics of unit cell structures helps in predicting and explaining the macroscopic properties of the material, such as its density and elasticity.

Corner Atoms Contribution

In a body-centered cubic unit cell, each corner atom contributes to multiple unit cells simultaneously. This is because each of the eight corner atoms is shared among eight separate unit cells.
To account for this, each corner atom is considered to contribute just one-eighth of an atom to the unit cell it is part of. Hence, with all eight corner atoms together in a BCC structure, their combined contribution boils down to one full atom ( \(8 \times \frac{1}{8} = 1\) atom) in that single cell. This calculation is crucial for determining the number of atoms truly present in the unit cell.

Center Atom Contribution

Contrary to corner atoms, the central atom in a body-centered cubic lattice is remarkably straightforward in its contribution. Unlike edges or faces, the center atom sits entirely within one unit cell. This means it contributes a full, undivided atom to that unit cell itself.
The concept becomes handy when trying to calculate the total number of atoms present in any arrangement. In the case of vanadium in a BCC lattice, the center atom adds one full atom outright, leading to a total of two atoms within the unit cell when combined with the corner atoms' contributions.