Density of Unit Cell

Introduction

The proportion of the mass of the unit cell to the volume of the unit cell gives a quantity called the density of the unit cell. The multiplication of the number of atoms and the mass of each atom in a unit cell is the mass of the unit cell.

The least volume-consuming and most basic repeating structure of solids are unit cells. It has application in the representation of the crystalline pattern of solids. And this visualization helps in the calculation of the density of unit cells too. A network is formed when a unit repeat and is called a lattice. And the particles present in the lattice are represented as points in space. It has several shapes, and it will depend on the angles between the edges and the length of the edges. The overall symmetry of solids lies in the type of unit cell. The density of a unit cell can be computed effortlessly by knowing the crystal lattice of unit cells.

What is a Unit Cell?

The structure of solids is a three-dimensional arrangement. And their pose a repeating unit for its arrangement. So, the smallest repeating unit of every structural unit of every solid is called the unit cell. The repetition of the unit cell results in the formation of a crystal. Based on the type of arrangement unit cells are of different types. Primitive and non-primitive unit cells are two categories.

When constituent particles are available only at the corners it is called a primitive unit cell. The presence of constituent particles in the corners and some other positions are non-primitive unit cells. And there are 6 parameters for the unit cell. They are three angles between them (𝛼, 𝛽, 𝑎𝑛𝑑 𝛾) and edges (𝑎, 𝑏, 𝑎𝑛𝑑 𝑐).

Benefits of Visualization

The visualization of the crystal structure of solids is important. They can be used to explain −

• The properties of solids.

• It will help to identify the stacking pattern.

• It will help in the identification of extended structures.

• It will give an idea about the ionic lattice of a solid.

• It will give the empirical formulas of solids.

• The knowledge of individual unit cells is obtained from the visualization.

What is Lattice?

The three-dimensional configuration of atoms present in a solid is the lattice. To reduce the total intermolecular energy, the items are arranged in a type of geometrical shape in solids. And is the lattice structure of solids. So, it is a kind of diagrammatic representation. And it includes atoms, ions, and molecules. The constituent particles are represented by points and are called the lattice point. The geometry of the crystal lattice is formed by the joining of these points with a straight line.

Bravais Lattices

In a crystal lattice if the surrounding lattice point is all same or all the items are identical it is called a Bravais lattice. In such crystals, the orientation arrangement of atoms is also the same. In 1948, Bravais depicted that there are 14 lattices are enough to discuss the crystals. And are Bravais lattice. There are 7 crystal systems based on the cell parameters in the Bravais lattice. And they are,

• Trigonal (or Rhombohedral)

• Tetragonal

• Triclinic

• Monoclinic

• Cubic

• Hexagonal

• Orthorhombic

Calculation of Density of Unit Cell

The density of the unit cell is the proportion of the mass of the unit cell and the volume of the unit cell. We can simply calculate it by the following equation.

$$\mathrm{Density\:,\rho\:=\:mass\:/\:volume}$$

The mass of a unit cell can be calculated by the equation,

$\mathrm{Mass\:of\:unit\:cell\:=\:number\:of\:atoms\:in\:unit\:cell\:\times\:mass\:of\:each\:atom\:}$

$$\mathrm{m\:=\:M/N_{A},\:Where\:M\:=\:molar\:mass}$$

The volume can be calculated by the knowledge of the edge length of unit cells. If a is the edge length the volume is,

$$\mathrm{V\:=\:a^{3}}$$

$\mathrm{S0\:,\:Density\:,\:\rho\:=\:z\:\times\:m/a^{3}\:=\:\frac{z\:\times\:M}{N_{A}a^{3}}}$

Example

A compound having a BCC crystal configuration has an atomic mass of 50 amu and an edge length is 290pm. Calculate the density in 𝑔𝑐𝑚−3 of the unit cell.

Answer: The equation that can be used for the calculation of density is, $\mathrm{\rho\:=\:z\:\times\:m/a^{3}\:=\:\frac{z\:\times\:M}{N_{A}a^{3}}}$

The number of atoms for BCC is 2(z).

$$\mathrm{\rho\:=\:\frac{2\:\times\:50}{6.022\:\times\:10^{23}\:\times\:(290\:\times\:10^{-10})^{3}}\:=\:6.81\:gcm^{-3}}$$

So, the density is $\mathrm{6.81\:gcm^{-3}}$

Significance of the Given Technique

By the expression of density calculation of unit cell, it has some significance. They are,

• It can be used for the calculation of edge length when all other factors are known.

• It can also be used for the calculation of a volume of a unit cell if the density is known.

• They can also be used for the calculation of the mass of the unit cell.

• It can be used for the calculation of densities of several crystals.

General Expression for Density of Unit Cells Derived for Various Cases

The general equation for the calculation of a density of a unit cell is,

$$\mathrm{\rho\:=\:z\:\times\:m/a^{3}}$$

What is the Planar Density of the (110) Plane in a Face-Centred Cubic? (FCC) Unit Cell

The number of atoms per unit area on a particular crystallographic plane is planar density. The planar density of FCC 110 is,

$\mathrm{\:\:\:\:\:Planar\:density\:=\:\frac{Number\:of\:atoms}{Area\:of\:plane}}$

$\mathrm{\:\:\:\:\:\:\:\:=\:0.177/\:(Radius\:of\:the\:constituent\:particle)^{2}}$

By knowing the radius of the constituent atom, we can calculate the planar density.

Primitive Unit Cell

When the constituent particles are present only at the intersections it is called a primitive unit cell. So, it contains only one lattice point. It is the basic unit cell of all other crystal structures since some additions to this unit cell led to the formation of other unit cells.

Body-Centred Cubic Unit Cell

It is a classification of non-primitive unit cells. All the eight corners of the cube consist of atoms and one in the middle leads to the construction of a Body-centered cubic unit cell. The coordination number of body-centred cubic unit cells is eight. The net total of atoms is two, one at the centre and one at the eight corners.

Mih.s29 , Lattice body centered cubic , CC BY-SA 4.0

Face-Centred Cubic Unit Cell

It Is also a type of non-primitive. In this kind of unit cell, there are eight atoms at the corners and the centre of all the faces of the cubic configuration. The coordination figure of this unit cell is 12. And the net total of atoms is four.

Mm5713 , Face Centred Cubic Lattice Correct Version , CC BY-SA 4.0 Fun Facts

About Unit Cells

Just like living organisms are made up of cells, all the crystals present in our surroundings are made up of the smallest repeating unit called the unit cell. So, it is the basic constituent of every crystal. The unit cell also represents the symmetry of the structures. They are arranged in a proper way to obtain the perfect shapes of every crystal.

Explanation of Relation between Lattice Constant and Density

Lattice constants are the parameters of all the crystal lattices. The edge length a, b, and c are the parameters. It is related to the density calculation in the following equation,

$$\mathrm{\rho\:=\:z\:\times\:m/(lattice\:constant)^{3}}$$

Diamond Cubic Structure

Diamond is a cubic structure with a Bravais lattice of the face-centred cube. And are made up of the element, carbon. So, there are eight atoms at the corners and the centre of all the faces of the cubic structure. So, the net total of atoms is four. Therefore, the Diamond contains a repeating unit of the face-centred cubic lattice.

Viktor Hakansson Ingre , Diamond cubic crystal structure , CC BY-SA 4.0

Conclusion

Unit cells are the smallest repeating unit of every crystal. Unit cells give symmetry and shapes to crystals. The three-dimensional arrangement of atoms leads to the formation of the lattice. There are two types of unit cells: primitive and non-primitive unit cells. There are six parameters in the unit cell: three angles and three edges. The density of every unit cell is calculated by using the equation, 𝜌 = 𝑧 × 𝑚/𝑎3. The lattice constant, a is related to the calculation of density. There are a total of seven crystal systems and are called Bravias lattice. The cubic structure of a diamond is the face-centred cubic lattice.

FAQs

1. What is an end-centred unit cell?

It is a type of non-primitive unit cell in which the particles are present at the 8 intersections and in the centre of two opposite faces. So, there is a net total of two atoms.

2. Is 𝑵𝒂𝑪𝒍 body-centred cubic?

NaCl is the face-centred cubic structure that contains four sodium and chlorine atoms.

3. Which is the example of hcp?

Many elemental metals have an hcp, hexagonal close-packed structure. For example, Titanium, Zinc, etc.

4. What is a simple cubic lattice?

A simple cubic lattice is the simplest unit cell. It contains only atoms at the corners of the cube. So, it is also called primitive cubic.

5. Is gold metal a simple cubic unit cell?

Gold metal is the face-centred cubic lattice. All the gold atoms are arranged as a repeating unit of FCC lattice.