cubic+unit+cells

The simplest repeating pattern in a crystal lattice is called a //unit cell//.

There are many classes of unit cells, but we will focus on three of the //cubic// type.

Below are some examples of cubic unit cells.




 * Cubic Unit Cells**

The cubic unit cell is one of the easiest to work with because it is simple to calculate the volume of a cube. In solid state studies, this volume is referred to as the unit cell volume, which is the cube of the //edge length//, **//a//**, of the unit cell.



Edge length, like atomic radius, is typically reported in units such as Angstroms, picometers, or centimeters.


 * Simple Cubic Unit Cell**

The //simple cubic// unit cell is one where each corner of the unit cell is occupied by a particle (atom, ion, molecule, etc.).

The edges of each face of the unit cell bisect the particle through its center (e.g. nucleus) such that only 1/8th of the volume of the particle is within the unit cell.



As you can see, the edge length //a// of the unit cell is equal to twice the radius of the particle:


 * a = 2 r**

The number of atoms or ions //inside// the unit cell will take up space, which is calculated by multiplying the number of corners (8) by the volume of each atom in that corner (1/8). This is called the //occupied volume// of the unit cell.


 * occupied volume of simple cubic: 8 corners x 1/8 volume = 1**

This volume can be calculated using the volume of a sphere if the radius of the atom or ion is known.


 * Body-centered cubic unit cells**

The body-centered cubic unit cell, or bcc cell, is sometimes referred to as the //cubic closest packed (CCP)// unit cell.

As the name implies, there is an atom (or ion) centered in the body of the unit cell:



Notice that each corner still contains 1/8th of a sphere.

To calculate occupied volume, add the 1 whole volume of the central atom or ion to that of the simple cubic occupied volume:


 * occupied volume of bcc unit cell: 8 corners x 1/8 volume = 1 + 1 center = 2**

The relationship between the edge length, //a//, and the radius of the sphere, r, is more complex than that of simple cubic.


 * a = 4 r / 3 1/2 or 2.31 r**

(You need to use Pythagorean Theorem a couple of times to derive this).

Nonetheless, due to the small gap between the atoms, the relationship is slightly larger in size (2.31 vs. 2).


 * Face-centered cubic unit cells**

The //face-centered cubic (fcc)// unit cell, as the name implies, has one sphere centered on each of the six faces of the cube of the unit cell:



The occupied volume would be calculated in this way:


 * occupied volume of fcc unit cell: 8 corners x 1/8 volume + 6 faces x 1/2 volume = 4**

Here again, the radii of the corner spheres do not touch. Using Pythagorean Theorem, the edge length to radius relationship is calculated to be:


 * a = r x 8 1/2, or 2.83 r**