Calculation of density of unit cell

The Unit Cell refers to the tiniest group of atoms with the general crystal’s symmetry  & from which the entire lattice can be built up by recurrence in 3 dimensions. The component particles in crystalline solids are arranged in a systematic and recurring pattern. A crystal is created on a lattice, which looks like a periodic 3-dimensional arrangement of points. M. A. Bravais demonstrated, in 1850, that identical points might be spatially arranged to generate 14 different sorts of patterns that are regular. Bravais lattices are a group of 14 space lattices.

A solid’s crystal lattice can be characterized using its unit cell. The crystal lattice comprises a vast quantity of unit cells, with a constituent atom occupying each lattice point. A 3-dimensional structure containing one or more atoms is referred to as a unit cell.

Crystal Lattice Characteristics

A crystal lattice has the following characteristics:

• A point in a lattice is known as a lattice point or site.
• A constituent particle, which could be an atom, a molecule, or an ion, is represented by each crystal lattice point. The geometry of the lattice is represented by lines connecting the points of the lattice.

The Formula of Density of Unit Cell

The volume of the unit cell can be calculated using the dimensions of a unit cell. For example, if an edge “a” contains a unit cell, the unit cell’s volume can be written as “a3.” 

The mass-to-volume ratio of a unit cell is known as its density. The unit cell’s mass is equal to the product of the number of atoms in a unit cell & each atom’s mass in a unit cell. The unit cell’s mass is proportional to the number of atoms in the cell multiplied by each atom’s mass, which is z m.

Where z denotes the number of atoms in a unit cell and m denotes the mass of each atom.

Calculation of density of unit cell

1. Cell of the Primitive Unit

In a primordial density of unit cell formula, there is only one atom. The density can be expressed as: 

ρ=(z×M)/(a3×NA)

∴ρ=1×M/a3×NA

2. Cubic Unit Cell centered on the body

In a body-centered density of unit cell formula, there are two atoms. The density can be expressed as:

ρ=(z×M)/(a3×NA)

∴ρ=2×M/a3×NA

3. Cubic Unit Cell with a Face

In a face-centered cubic unit cell, there are four atoms. The density of unit cell formula can be expressed as:

ρ=(z×M)/(a3×NA)

∴ρ=4×M/a3×NA

The density of Crystal Lattice Formula

The first thing to know about cubic crystals is that they are made up of a crystal lattice or a space lattice structure. A unit cell is the most fundamental building block of the crystalline solid . As we’ve seen earlier, a space lattice is formed out of a repeating pattern of unit cells.

As a result, finding the density of a unit cell is equivalent to finding the density of the cubic crystal itself. Let’s look at how to compute a cubic crystal cell’s density with this in mind.

A Unit Cell’s Calculations

A unit cell, as we all know, has a cubic structure. It has one, two, or four atoms distributed over the lattice. We can now calculate the density of a unit cell using geometry, some simple computations, and some characteristics of this cubic structure. Let’s start with the most basic formula for determining a solid’s density. This is the formula.

Mass/Volume = Density

A Unit Cell’s density will be

D = Unit Cell Mass/Unit Cell Volume

Now, to compute the mass of a single unit cell, we add the masses of all the atoms in that cell. The amount of atoms in a cell is determined by the cell type. So we multiply the atoms’ number “n” by the mass of each atom “m” to get the mass of a unit cell.

However, the topic of an atom’s mass is still open. This can be expressed in terms of its Avogadro Number (NA), equal to the number of units in one mole of any substance multiplied by the molar mass of an atom. As a result, the mass of an atom can be estimated as follows:

Avogadro’s formula for calculating the mass of an atom is MolarMass/formula Avogadro’s for calculating the mass.

M/NA is the mass of an atom. As a result, the formula for Unit Cell Mass is as follows:

n M/NA = Mass of Unit Cell

A unit cell’s volume

The unit cell is a cubic structure, as we well know. Assume that the length of the cube’s side is “a” and that the cube’s volume is the cube of the length of the side.

Unit Cell Volume = a3

Unit Cell Density is a measure of how dense a unit cell is.

We’ve finally arrived at the density of the unit cell formula.

Mass of Unit Cell/ Volume of Unit Cell

nM/a3NA = Density of a Unit Cell

Conclusion

The unit cell is the most basic & least volume-consuming recurring structure of any material. Its purpose is to visibly simplify the crystalline patterns that solids form. Whenever the unit cell repeats itself, the network is termed a lattice. The unit cell can take on a number of shapes depending on the angels among cell edges & their corresponding edges.

It is the fundamental building block with a unique atomic configuration.