Question

Packing fraction in FCC lattice is (a) \(\frac{1}{6} \pi\) (b) \(\frac{\sqrt{2}}{6} \pi\) (c) \(\frac{\sqrt{3}}{8} \pi\) (d) \(\frac{\sqrt{2}}{3} \pi\)

Short answer

\(\frac{\sqrt{2}}{6} \pi\)

Step-by-step solution

Define the concept of packing fraction

Packing fraction, in the context of crystal lattices, is the fraction of the volume of the unit cell that is occupied by the atoms. For a Face-Centered Cubic (FCC) lattice, each unit cell has 8 corner atoms and 6 face-centered atoms. However, each corner atom is shared between 8 unit cells and each face-centered atom is shared between 2 unit cells. This effectively gives us 4 whole atoms per unit cell.

Calculate the volume occupied by atoms

Assuming each atom in the FCC lattice is a sphere with radius r, the volume of one atom would be \( \frac{4}{3} \pi r^3 \). Since there are effectively 4 atoms per FCC unit cell, the total volume occupied by the atoms is \( 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \).

Calculate the volume of the FCC unit cell

The side of the FCC unit cell is \( 2\sqrt{2}r \) because the diagonal across the face of the unit cell (which is \( \sqrt{2} \times \text{side length} \)) passes through two radii. So, the volume of the unit cell is \( (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \).

Compute the packing fraction

The packing fraction is the ratio of the volume occupied by atoms to the volume of the unit cell: \[ \text{Packing fraction} = \frac{\text{Volume occupied by atoms}}{\text{Volume of the unit cell}} = \frac{\frac{16}{3} \pi r^3}{16\sqrt{2}r^3} = \frac{\pi}{3\sqrt{2}} \]. Multiplying both numerator and denominator by \(\sqrt{2}\) gives us: \[ \frac{\sqrt{2} \pi}{6} \]

Key concepts

Crystal Lattice Structures

Imagine solids composed of a repeating arrangement of atoms, ions, or molecules in three-dimensional space; this is what we refer to as a crystal lattice structure. Each point in this lattice represents the position of a particle and the entire arrangement is orderly and geometrically fixed.

There are different types of crystal lattices, such as simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC). These structures differ in how particles are arranged within the unit cell, which is the basic repeating unit that defines the crystal structure. For example, in an FCC lattice, atoms are located at each of the corners and at the centers of all the cube faces.

Understanding the arrangement of atoms within these lattices is key to determining the properties of materials, such as density, melting point, and how they interact with light. Students often construct models or use diagrams to visualize the three-dimensional structures to better understand the spatial relations between atoms.

Unit Cell Volume Calculation

When we talk about the unit cell volume calculation, we're essentially aiming to measure the space that the unit cell occupies in a crystal lattice. This value is crucial as it helps understand the density and packing efficiency of the lattice.

To calculate the unit cell volume, we first need to know the geometry of the unit cell. For instance, if we have a cubic unit cell, the volume is simply the cube of the edge length. In more complex lattices like the FCC, additional geometric analysis is required to relate the edge length to the radii of atoms placed on the lattice points.

Relation with Atom Radii

Once we have the edge length, we can take advantage of geometry to find the volume. In an FCC lattice, as we've seen from the solution, the diagonal passing through the face of the cube enables us to express the cell's side length in terms of the atomic radius. It's this relationship that often provides insight into the compactness of the atomic arrangement within the cell.

Face-Centered Cubic (FCC)

Delving deeper into the Face-Centered Cubic (FCC) lattice, we find it is one of the most tightly packed crystal structures, alongside the hexagonal close-packed (HCP) structure. In an FCC lattice, atoms are not only at each corner of the cube but also in the center of each face. This results in a total coordination number of 12, meaning each atom touches 12 others.

Since the atoms at the corners and the centers of the faces are shared among adjacent unit cells, the effective number of atoms per unit cell needs to be calculated to understand the structure's density and packing efficiency. This is crucial in materials science, influencing the material's strength and other physical properties.

Packing Efficiency and Material Properties

Metals like copper, aluminum, and silver crystallize in an FCC structure, exhibiting high packing efficiency and therefore contributing to their malleability and ductility. The closer atoms are packed, the easier they can slide past each other, which is essential for understanding the mechanical behavior of metals under various stresses.