Question

The number of atoms per unit cell in a BCC, a FCC and a simple cubic cell are respectively (a) \(1,4,2\) (b) \(2,4,1\) (c) \(4,1,2\) (d) \(4,2,2\)

Short answer

The correct answer is (b) \(2, 4, 1\).

Step-by-step solution

Understand the Types of Unit Cells

In crystal structures, a unit cell is the smallest repeating unit that shows the full symmetry of the arrangement of atoms. Common unit cells include Body-Centered Cubic (BCC), Face-Centered Cubic (FCC), and Simple Cubic (SC). We'll calculate the number of atoms in each of these unit cells.

Calculate Atoms in a Body-Centered Cubic (BCC) Cell

In a BCC structure, there is one atom at each of the eight corners of the cube and one atom in the center. At each corner, atoms are shared by 8 adjacent unit cells. Therefore, the effective number of atoms per corner is \( \frac{1}{8} \times 8 = 1 \). The center atom belongs entirely to that unit cell. Therefore, total atoms in BCC = 1 (from corners) + 1 (from center) = 2.

Calculate Atoms in a Face-Centered Cubic (FCC) Cell

In an FCC structure, in addition to the 8 corner atoms, each face of the cube has an atom. Atoms on the faces are shared by 2 adjacent unit cells. Therefore, the effective number of atoms per face is \( \frac{1}{2} \times 6 = 3 \) as there are 6 faces. Adding these to the corners, total atoms in FCC = 1 (from corners) + 3 (from faces) = 4.

Calculate Atoms in a Simple Cubic (SC) Cell

In a simple cubic structure, there is one atom at each of the 8 corners. The corner atoms each contribute \( \frac{1}{8} \) of an atom to the unit cell for a total of \( 8 \times \frac{1}{8} = 1 \) atom per unit cell.

Match with Provided Options

We calculated that a BCC cell has 2 atoms, an FCC cell has 4 atoms, and a Simple Cubic cell has 1 atom. This matches with option (b), which states \(2, 4, 1\) respectively.

Key concepts

Unit Cell

A unit cell is the fundamental building block of a crystal structure. It provides a repeating pattern that extends in all three dimensions to form the entire crystal lattice. Think of it as a small box that shows how atoms are arranged in space. The arrangement and number of atoms within a unit cell determine the properties of the crystal.
Unit cells come in several types, with the most common being Simple Cubic, Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC). Each type has a unique way of packing atoms, which defines how dense and stable the crystal structure can be.

• A Simple Cubic unit cell has atoms only at the corners.
• A Body-Centered Cubic has an additional atom at the center of the cube.
• A Face-Centered Cubic features atoms at the center of each face.

These arrangements influence how materials conduct electricity, resist deformation, and more, making an understanding of unit cells essential in material science and engineering.

Body-Centered Cubic Structure

The Body-Centered Cubic (BCC) structure is one of the most important types of unit cells in solid-state chemistry. In a BCC arrangement, each unit cell has atoms located at the corners of a cube, but it also includes a single atom at the very center of the cube. This central atom contacts all the corner atoms, giving the structure its name.
To count the number of atoms in a BCC unit cell:

• Each corner atom is shared among 8 neighboring unit cells, so their contribution is \( \frac{1}{8} \) per corner. With 8 corners, that amounts to 1 atom.
• The center atom is entirely within the unit cell, contributing 1 full atom.

Thus, a BCC unit cell contains 2 atoms in total.
The BCC structure is typical of metals like iron and tungsten, providing a strong yet flexible arrangement that’s crucial for construction materials.

Face-Centered Cubic Structure

The Face-Centered Cubic (FCC) structure packs atoms very efficiently, with each unit cell containing atoms at all the corners and the centers of each face of the cube. This arrangement provides a highly dense packing.
Calculating the number of atoms in an FCC unit cell involves:

• 8 corner atoms, each shared by 8 unit cells, contributing \( \frac{1}{8} \) of an atom per corner. Combined, these corners contribute 1 atom.
• 6 face-centered atoms, with each shared by 2 unit cells, contributing \( \frac{1}{2} \) of an atom per face. Therefore, the faces contribute 3 atoms.

Adding these contributions together, an FCC unit cell contains 4 atoms.
Materials such as aluminum, copper, and gold utilize the FCC structure, which gives them their extraordinary ductility and high thermal and electrical conductivity.

Simple Cubic Structure

The Simple Cubic (SC) structure is the most basic arrangement of atoms in crystalline structures. It consists solely of atoms at each of the eight corners of the cube. The simple cubic structure is rarely found in nature because it doesn't pack atoms as densely as other structures do, but it's easier to visualize and understand.
Each of the 8 corner atoms in a simple cubic unit cell is shared by 8 neighboring cells. Therefore, the contribution of each corner atom is \( \frac{1}{8} \). This means that for all 8 corners:

• Each contributes \( \frac{1}{8} \), totaling to 1 atom when all 8 corners are considered.

Thus, a simple cubic unit cell contains 1 atom.
Because of its inefficiency in space, the simple cubic structure isn't common among materials, but it serves as a fundamental concept for learning about more complex arrangements.