Question

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

Short answer

\(\frac{1}{6} \pi\)

Step-by-step solution

Understanding the Concept of Packing Fraction

The packing fraction is defined as the fraction of volume in a crystal structure that is occupied by the atoms themselves. In a simple cubic lattice, atoms are arranged in such a way that each atom sits on a corner of a cube and touches the other atoms only along the corners.

Determining the Volume of Sphere in the Cube

The atoms in a simple cubic lattice can be assumed to be hard spheres. In a simple cubic unit cell, there is one atom per unit cell and each corner of the cube is shared by eight different unit cells. Therefore, the volume of one atom (a sphere) is \(\frac{1}{8}\) of the volume of a sphere with radius R in the cell, which can be calculated by the formula for the volume of a sphere: \(\frac{4}{3} \pi R^3\).

Calculating the Volume of the Cube

The edge length of the cube, a, is equal to two times the radius of the sphere R (since the sphere's diameter is equal to the edge length of the cube). Therefore, the volume of the cube is \(a^3 = (2R)^3 = 8R^3\).

Calculating the Packing Fraction

The packing fraction is the ratio of the volume occupied by the spheres (atoms) to the volume of the cube. The volume occupied by one atom in a simple cubic lattice is \(\frac{1}{8}\) of the volume of the whole sphere. Thus, the packing fraction is \(\frac{\frac{1}{8} \times \frac{4}{3} \pi R^3}{8R^3} = \frac{\pi}{6}\).

Key concepts

Crystal Structure

When we dive into the microscopic realm of materials science, we encounter the intriguing world of crystal structures. This is the intricate three-dimensional arrangement of atoms, ions, or molecules in a crystalline material.

The simplest type is the simple cubic lattice, which is characterized by atoms located at each corner of a cube-shaped unit cell. Imagine it like a three-dimensional chessboard, where each square is a part of a much bigger and perfectly ordered structure. Each cube-shaped unit cell in this lattice type repeats itself in a uniform pattern across three dimensions, creating the solid crystal.

Now, just because the arrangement is 'simple,' don't underestimate its importance. The simple cubic structure, while not the most densely packed, provides a foundational understanding of how matter is organized at the atomic level and influences the physical properties of materials.

Volume of a Sphere

Understanding the volume of a sphere is crucial when we think about atoms in crystal structures, as atoms are often approximated as spherical objects. The formula to calculate a sphere's volume, \( \frac{4}{3} \pi R^3 \), might look intimidating at first glance, but it's rooted in basic geometry and gives us essential insight into how much space an atom occupies.

In this formula, \( R \) represents the radius of the sphere – the distance from the very center of the sphere to its surface. The pi symbol \( (\pi) \) is the ratio of the circumference of a circle to its diameter, a constant roughly equal to 3.14159. The beautiful part of this formula is that it encapsulates a fundamental property of all spheres, regardless of their size: the way they occupy space.

Unit Cell

The concept of a unit cell is the cornerstone of understanding crystalline structures. Simply put, the unit cell is the smallest portion of a crystal lattice that repeats in three-dimensional space to form the entire lattice. Think of it as the 'building block' of the crystal. In the case of a simple cubic lattice, the unit cell is, well, a cube.

Every corner of this cube 'shares' its atoms with adjacent unit cells; this is critical when calculating the 'true' volume an atom occupies within a crystal. A single unit cell in a simple cubic structure contains only a fraction of each of its corner atoms. Only \( \frac{1}{8} \) of an atom belongs to one specific unit cell, as it is shared equally among eight surrounding cells.

This sharing influences calculated quantities such as the packing fraction, which essentially tells us how much space within the cell is 'filled' by atoms. In essence, the unit cell is a window into the symmetry and efficiency with which atoms are packed in solid materials, influencing their properties and behaviors.