Animated Solution for Chemistry - Solid State: Diamond has a three dimensional structure of C atoms formed by covalent bonds. The structure of diamond has face centred cubic lattice, where 50% of the tetrahedral voids are also occupied by carbon atoms. The number of carbon atoms present per unit cell of diamond is …… .

Decoding the FCC Lattice

Imagine holding a sparkling diamond. Beyond its brilliant exterior lies a microscopic world of incredible symmetry and strength. The secret to diamond's legendary hardness is hidden in its crystal lattice. At its core, diamond crystallizes in a Face-Centered Cubic (FCC) lattice.

Let's break down what this means. In an FCC unit cell, carbon atoms are positioned at all eight corners of the cube and exactly in the center of all six faces. However, these atoms are shared with neighboring unit cells. A corner atom is shared by eight adjacent cubes, contributing only $\frac{1}{8}$ to our specific unit cell. A face-centered atom is shared by two cubes, contributing $\frac{1}{2}$.

So, the primary FCC lattice provides us with exactly $4$ effective carbon atoms.

But the story of diamond doesn't end with just the lattice points. Inside this FCC structure, there are empty pockets of space known as tetrahedral voids. These voids are located along the body diagonals of the cube, exactly one-fourth of the distance from each corner.

A fundamental rule in solid-state chemistry tells us that the number of tetrahedral voids is always twice the effective number of atoms forming the lattice. Since our FCC lattice has $4$ atoms, we can easily calculate the total number of tetrahedral voids:

This means there are $8$ potential parking spots for extra atoms inside the unit cell.

Here is the fascinating catch that makes diamond unique. If all $8$ tetrahedral voids were filled with carbon atoms, they would be packed too closely together, leading to massive steric repulsion and instability. Nature solves this by occupying only $50\%$ of these available voids.

By filling only half the voids, the carbon atoms arrange themselves alternately, maximizing the distance between them and minimizing repulsion. Let's calculate how many voids are actually occupied:

So, exactly $4$ carbon atoms sit snugly inside these tetrahedral voids.

Now, we simply bring everything together to find the total number of carbon atoms in a single unit cell of diamond. We take the atoms from the main FCC lattice and add the atoms residing in the tetrahedral voids.

There we have it! A single unit cell of diamond contains exactly $8$ carbon atoms. Interestingly, because it only fills half of its voids, diamond has a relatively low packing efficiency of about $34\%$. It is an "open" structure, yet its strong, directional covalent bonds make it the hardest known natural material on Earth.