The Density of a Unit Cell

Do you want to learn how to calculate the total density of a unit cell calculator? If yes, then you are in the right place. Here, we will learn to calculate the density of an fcc unit cell. Before you calculate the density of an fcc unit cell, you must know what a unit cell is. A unit cell can be described as the smallest group of atoms that have the overall symmetry of a crystal. It is a structure that can be used to build an entire lattice by repetition in three dimensions.

It is important to remember that crystalline solids also exhibit a regular and repeating pattern of constituent particles. Before you calculate the density of a unit cell, you must also be able to visualize its complete structure of it. The image of a unit cell is given below for your reference.

(Image to be added soon)

Benefits of Visualization

This visualization will go a long way in helping students calculate the density of a unit cell. Students can also use this technique to calculate the density of body-centered cubic unit cells. On the same note, students should also keep in mind that the diagrammatic representation of three-dimensional arrangements of constituent particles of a crystal is known as a crystal lattice. In a crystal lattice, each particle is depicted as a particular point in space.

What is Lattice?

Before moving on to the next topic of how to calculate the density of a body-centered cubic unit cell let us understand what a lattice is. According to experts, lattice can be described as a framework that resembles a three-dimensional periodic array of points. It is this framework on which an entire crystal is built.

Bravais Lattices

Every student who wishes to know the answer to the question of how to calculate the density of empty space within a unit cell has to also learn the theory of this topic. It all started in 1850 when M. A. Bravais showed the world that identical points can also be arranged spatially. This arrangement would produce 14 different types of regular patterns. Students might also be interested to know that these 14 space lattices are known as Bravais lattices.

Calculation of Density of Unit Cell

To further continue the discussion on how to calculate the density of empty space within a unit cell, it is also vital for students to note that a crystal lattice of a solid could also be described in terms of its unit cells.

As mentioned above, a crystal lattice consists of a very large number of unit cells, and every lattice point is occupied by one constituent particle. Hence, the density of unit cell fcc can be viewed as a three-dimensional structure that contains one or more atoms. All of this information can be used to determine the density of unit cell fcc. To understand all of these how to find the density of a one-unit cell clearly.

Example

Let’s take an example. Let’s assume that an individual has a unit cell that has an edge ‘a.’ The volume of that unit cell is ‘v,’ and the density of the unit cell is given as the ratio of mass and volume of the unit cell.

Further, the mass of the unit cell is equal to the product of the number of total atoms in the unit cell and the mass of every atom in the unit cell. Now, one has to learn how to find the density of a one-unit cell to arrive at the final answer. Let us begin the answer by noting down everything that we already know. 

Mass of unit cell = number of total atoms in unit cell x mass of every atom = z x m

In this equation for finding out the density of unit cell material science, ‘z’ is the total number of atoms in a unit cell, and ‘m’ is the mass of every atom. Students must also be familiar with the fact that the mass of an atom can be calculated with the help of Avogadro number and molar mass, that is, M / Na

Here, ‘M’ is the molar mass, and ‘Na’ is the Avogadro’s number. Further, the volume of the unit cell or ‘V’ = A3

Hence, the density of the unit cell = Mass of the unit cell /  Volume of the unit cell

This further means that density of the unit cell is equal to m/V = z x m / A3 = z x M / A3 x Na

It can be concluded that if one knows the total number of atoms in a unit cell, edge length, and molar mass, then one can calculate the density of the unit cell.

Significance of the Given Technique

This same technique can be used to calculate the density of NaCl unit cells and the density of polonium unit cells. We also learned that the density of a unit cell equals extended lattice, and it is easy to determine the density given the length of the unit cell.

General Expression for Density of Unit Cells Derived for Various Cases

Till now, we have learned how to calculate the density of a NaCl unit cell, the density of a polonium unit cell, the fact that the density of a unit cell equals an extended lattice, and it is relatively easy to determine the density given the length of a unit cell.

What is the Planar Density of the (101) Plane in a Face-Centered Cubic (fcc) Unit Cell

Now, let us go through the answer to the question of what is the planar density of the (101) plane in a face-centered cubic (fcc) unit cell. Students should also be familiar with the general expressions used for the density of unit cells. These expressions are derived for various cases. And to help students to understand this topic better, we have formulated a list of important expressions. The list is given below.

Primitive Unit Cell

In a primitive unit cell, the total number of atoms in a unit cell is equal to one. This means that the density of a primitive unit cell is equal to 1 x M / A3 x Na.

Body-Centered Cubic Unit Cell

When it comes to a body-centered cubic unit cell, then the total number of atoms in a unit cell is two. This means that the value of z is two. Hence, the density of a body-centered cubic unit cell is equal to 2 x M / A3 x Na.

Face-Centered Cubic Unit Cell

The total number of atoms present in a face-centered cubic unit cell is four. Hence, the density of a face-centered cubic unit cell is 4 x M / A3 x Na.

Fun Facts about Unit Cells

Did you know that primitive cells only have lattice points at the corners of their cell? In contrast to primitive cells, unit cells, instead, have lattice points at their edges, face centers, and body centers! This means that an individual can examine the entire crystal by simply investigating a single unit cell.

Explanation of Relation between Lattice Constant and Density 

• Consider 'a' as the lattice constant of the cubic crystal.

• We know that the density of the crystal is represented by 'P'.

• V = a³ represents the volume of the unit cell (cubic crystal).

• n = Total number of atoms / unit cell

• M represents the material's atomic weight.

• N is the Avogadro constant in this equation.

Derivation of Formula for Density of Unit Cell

Now, the mass of 1 atom is equal to M/N.

For n number of molecules, the mass of the unit cell will be equal to nM/N. ….(i)

Also, density = P = mass per unit volume. So, P = m/V.

Now, P = m/a³ ….. (ii)

and here, m= mass

So, from (i) and (ii),  P = nM / N a³.

Relation between Lattice Constant and Density 

Density = Number of atoms per unit cell Atomic weight / Avogadro number x (Lattice constant)^3.

Diamond Cubic Structure 

• In a diamond cubic structure, eight corner atoms are present along with six face-centered atoms and four more atoms.

• The number of atoms contributed to the unit cell (by the corner atom) is ⅛ × 8 = 1.

• Similarly, The number of atoms contributed to the unit cell (by the face-centered atom) is ½ × 6 = 3.

• Here, a diamond cubic structure is a repeating pattern consisting of 8 atoms. A diamond cubic structure is considered to be FCC.

• The materials that are only made out of carbon are an example of their type of cubic structure. It can be better understood by the concepts of Crystallography.