Question

What is the coordination number of each sphere in (a) a simple cubic lattice, (b) a body-centered cubic lattice, and (c) a face- centered cubic lattice? Assume the spheres to be of equal size.

Short answer

The coordination numbers for a simple cubic lattice, a body-centered cubic lattice, and a face-centered cubic lattice are 6, 8, and 12 respectively.

Step-by-step solution

Coordination Number in Simple Cubic Lattice

In a simple cubic lattice, each sphere (or atom) is adjacent to six others: one each in left and right, up and down, and front and back. Thus, the coordination number for a simple cubic lattice is 6.

Coordination Number in Body-Centered Cubic Lattice

A body-centered cubic lattice has an additional sphere in the center of the cubic unit. This sphere is in touch with eight others which are at the corners of the cube, so the coordination number here is 8.

Coordination Number in Face-Centered Cubic Lattice

In a face-centered cubic lattice, each sphere is in contact with those in the center of all six faces, plus the twelve spheres that enclose the corners of the faces. This gives a coordination number of 12.

Key concepts

Simple Cubic Lattice

A simple cubic lattice is one of the most straightforward crystal structures to understand. Imagine stacking layers of spheres in a box-like pattern. In this type of lattice, each sphere is connected to six other spheres, which we call the coordination number. This number comes from having one sphere each in front, behind, to the left, to the right, above, and below a given sphere.

• Easy to visualize: It's like stacking oranges in a regular box format where every orange touches six others.
• Limited neighboring points: The structure is simple, but each sphere only touches a few others.
• Low packing efficiency: This structure is not very dense; it uses space inefficiently compared to other types of lattices.

Understanding these characteristics can help in visualizing how atoms in a simple cubic lattice interact with each other in real materials.

Body-Centered Cubic Lattice

The body-centered cubic lattice introduces a bit more complexity. Picture a cube with an extra sphere right in the center. This central sphere makes contact with eight corner spheres of the cube, forming a three-dimensional connection. Hence, the coordination number for a body-centered cubic lattice is eight.

• Added central sphere: This unique feature enhances the structural stability compared to a simple cubic lattice.
• Increased density: The additional sphere in the center means this lattice is more tightly packed.
• Common in metals: Many metals, like iron and tungsten, naturally crystallize in this pattern due to its stability.

The increase in the coordination number and packing efficiency makes this lattice an interesting study in material sciences.

Face-Centered Cubic Lattice

The face-centered cubic lattice takes the concept of packing efficiency to an advanced level. Here, each cubic face features an additional sphere in the center, alongside the eight corner spheres. This arrangement allows each sphere to be in contact with 12 others, which is why the coordination number is 12.

• High coordination: Having spheres at the face and corners enables topomechanical stability and maximizes space usage.
• Optimal packing: This is the most space-efficient of the basic cubic lattices, often found in metals like aluminum and copper.
• Close-packed structure: This is one of the densest structures, creating a robust crystal lattice.

Recognizing these qualities helps in understanding why materials with face-centered cubic lattices have specific physical properties such as high ductility and strength.