Question

The closet distance between two atoms (in terms of edge length) would be highest for which of unit cell, assuming the edge length of each unit cell of be 'a' (a) FCC unit cell (b) BCC unit cell (c) diamond unit cell (d) primitive cubic cell

Short answer

The closet distance is highest for the Primitive Cubic unit cell.

Step-by-step solution

Understanding Unit Cell Types

There are four unit cell types given: Face-Centered Cubic (FCC), Body-Centered Cubic (BCC), Diamond Cubic, and Primitive Cubic. Each of these has a different arrangement and coordination of atoms, affecting the closest distance between two atoms.

Closest Distance in FCC

In an FCC unit cell, atoms are located at each corner and at the centers of all the faces. The closest distance between two atoms in an FCC lattice is half of the face diagonal, which is \( \frac{a}{\sqrt{2}} \), where \( a \) is the edge length.

Closest Distance in BCC

In a BCC unit cell, atoms are positioned at each corner and a single atom is at the center of the cube. The closest distance between atoms in a BCC lattice is along the body diagonal, calculated as \( \frac{a\sqrt{3}}{2} - a \cdot \frac{\sqrt{3}}{4} \), which simplifies to roughly \( \frac{a\sqrt{3}}{2} \).

Closest Distance in Diamond Cubic

A Diamond cubic unit cell has atoms arranged such that each atom forms a tetrahedral coordination. The closest distance between atoms is along the tetrahedral bond, given by \( \frac{a}{4} \cdot \sqrt{3} \), which is smaller than in BCC and FCC.

Closest Distance in Primitive Cubic

In a Primitive Cubic unit cell, atoms are only at the corners of the cube. The closest distance is simply the edge length \( a \), as atoms touch along the edges of the cube.

Comparing Distances

After calculating the closest atomic distances for each unit cell, we compare them: FCC \( \frac{a}{\sqrt{2}} \), BCC \( \frac{a\sqrt{3}}{2} \), Diamond \( \frac{a\sqrt{3}}{4} \), and Primitive Cubic \( a \). Simplifying numerical evaluations show that the Primitive Cubic gives the highest distance \( a \).

Key concepts

Unit Cell Types

In the world of crystals and solids, understanding unit cells is crucial. A unit cell is the smallest repetitive structure of a solid that describes the entire structure by repeating in three-dimensional space.
There are different types of unit cells, each with a unique atomic arrangement:

• Face-Centered Cubic (FCC)
• Body-Centered Cubic (BCC)
• Diamond Cubic
• Primitive Cubic

Each type affects the properties of the material, including density, coordination number, and how atoms fit together.
By studying unit cells, scientists can predict material properties. This knowledge is vital in fields like material science and engineering.

Coordination of Atoms

In a crystal, atoms do not float haphazardly but are precisely ordered. This leads us to the concept of coordination of atoms, which refers to how an atom is surrounded by immediate neighboring atoms. It plays a vital role in defining the stability and properties of the structure.
The coordination number is specific to each unit cell:

• In a Primitive Cubic unit cell, each atom touches 6 neighbors.
• FCC unit cells have a coordination number of 12, maximizing atomic packing.
• BCC unit cells coordinate with 8 neighboring atoms.
• While in a Diamond Cubic structure, tetrahedral coordination is typical, with each atom connected to 4 others.

This coordination impacts the physical and chemical properties of the material.

Primitive Cubic Unit Cell

The Primitive Cubic unit cell is the simplest and least complex structural arrangement. It consists of atoms positioned solely at the corners of the cube. This simplicity results in a limited efficiency in terms of space use and atomic packing.
The main characteristics of a Primitive Cubic cell are:

• Coordination number is 6, relatively low among unit cell types.
• The closest distance between two atoms is equal to the edge length, denoted as \( a \).

Because of its simple design, it is a less common natural structure but important for understanding fundamental crystallography.

Face-Centered Cubic (FCC)

Face-Centered Cubic, or FCC, is a prevalent and efficient unit cell type. In this configuration, atoms are at each cube corner and the centers of each face, optimizing space usage and coordination.
Some key aspects of FCC:

• It has the highest coordination number of 12.
• The closest approach between any two atoms is along the face diagonal, given by \( \frac{a}{\sqrt{2}} \).

Owing to its high packing efficiency, FCC structures are common in nature, appearing in metals like copper, aluminum, and gold.

Body-Centered Cubic (BCC)

The Body-Centered Cubic unit cell presents a distinct structure where atoms occupy each corner and one in the center of the cube. This design offers a unique balance between simplicity and packing efficiency.
Important traits of BCC:

• The coordination number is 8, providing moderate atomic stability.
• The closest distance between atoms is calculated on the body diagonal, approximately \( \frac{a\sqrt{3}}{2} \).

BCC is commonly found in metals like iron and chromium, known for their strength due to this configuration.

Diamond Cubic Structure

The Diamond Cubic structure is famous for its use in diamonds and semiconductors. It features a complex yet fascinating atomic arrangement where atoms align in a tetrahedral pattern.
This structure has several defining characteristics:

• Each atom in the lattice is coordinated with 4 others, forming a simple tetrahedron.
• The closest distance between atoms is along a bond, calculated as \( \frac{a\sqrt{3}}{4} \).

This unique structure contributes to the exceptional hardness of diamonds and influences electrical properties in semiconductors like silicon and germanium.