Unit Cell: Lattice Parameters & Cubic Structures

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  • 0:04 Unit Cells & Lattice…
  • 0:58 Cubic Unit Cells
  • 1:29 Simple Cubic Structure
  • 2:31 Body-Centered Cubic Structure
  • 3:26 Face-Centered Cubic Structure
  • 4:42 Lesson Summary
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Lesson Transcript
Instructor: Stephanie Bryan

Stephanie has a master's degree in Physical Chemistry and teaches college level chemistry and physics.

In this lesson, we describe the lattice parameters of a unit cell and detail three cubic structures, their lattice parameters and equations that describe their geometry.

Unit Cells and Lattice Parameters

Crystal structures are made up of repeating units of atoms in well defined locations in a lattice. In order to describe these structures, it is useful to define the smallest unit of this structure. This is called the unit cell. The unit cell is the smallest unit of a crystal structure that can be used to tile space and make the larger macroscopic structure.

There are many shapes and patterns of unit cells. To describe these shapes, we use lattice parameters, or variables that describe the orientation of the unit cell. The typical lattice parameters that are used are the length of each side, which are typically labelled a, b, and c, and the angles between these sides, which are typically labelled α, β, and γ. In this lesson we are going to discuss three cubic unit cells in detail. A cubic unit cell is exactly the way it sounds: its basic outer structure is a cube.

Cubic Unit Cells

There are three different types of cubic unit cells we will discuss: the simple cubic, the body-centered cubic, and the face-centered cubic structure. While all three of these are cubes, they differ in the way the atoms are arranged inside the cube and therefore have different equations that describe their parameters. However, because all these structures are cubic, the outside edges are all equal. We will call this length variable a, and all angles in the unit cell are 90°. Therefore, the volume of the unit cell for all three will be a3.

Simple Cubic Structure (SC)

We will start with the simple cubic structure because it is, as it states, the most simple.

Cubic crystal shape

In the simple cubic structure, an atom exists at each corner of the cube. If we were counting atoms inside the cube we would see that each of these corner atoms only actually has 1/8 of its structure in the actual cube. There are 8 corners, so the simple cubic structure contains 8*(1/8) or 1 atom per unit cell. The radius of an atom is related to the length of the unit cell side by the following equation:


Because there is only one atom per unit cell, and an atom has a volume equal to (4/3)πr3, we can also determine the packing efficiency:

packing efficiency simple cubic

Simple cubic structures have a packing efficiency of 52%. This means that 52% of the structure is made up of atoms, and the remaining 48% is empty space.

Body-Centered Cubic Structure (BCC)

The body-centered cubic structure is similar to the simple cubic; however, it has an atom at the center of the cell as well as the corners.

body centered cubic structure

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