Animated Solution for Chemistry - States of Matter: A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is '$a$', the closest approach between two atoms in metallic crystal will be

Imagine you are looking at a single face of a Face-Centered Cubic (FCC) unit cell. In this arrangement, atoms are located at all eight corners of the cube, and there is an additional atom sitting right in the center of each of the six faces.

Now, a common misconception is that the atoms touch each other along the edges of the cube. However, because of the atom sitting in the face center, the corner atoms are pushed slightly apart. They do not touch along the edge $a$. Instead, they touch diagonally across the face.

Let's focus on that face diagonal. If the edge length of our cubic unit cell is $a$, we can easily find the length of the face diagonal using the Pythagoras theorem. The diagonal forms the hypotenuse of a right-angled triangle with two edges as its base and perpendicular.

Now, let's look at this same diagonal from the perspective of the atoms themselves. If we trace a line from one corner, through the face center, to the opposite corner, what do we cross? We cross the radius $r$ of the first corner atom, the full diameter $2r$ of the face-centered atom, and the radius $r$ of the second corner atom.

Since both expressions represent the exact same physical length, we can equate them to find the relationship between the atomic radius $r$ and the edge length $a$.

The question asks for the closest approach between two atoms. In any crystal lattice, the closest approach is simply the distance between the centers of two atoms that are physically touching each other. Since the corner atom and the face-centered atom are touching, the distance between their centers is exactly $2r$.

Let's substitute the value of $r$ we just derived into our expression for the closest approach $d$:

By rationalizing the denominator, we arrive at our final, elegant result:

This fundamental geometric relationship is the key to unlocking packing efficiency and density calculations for all FCC metals!