Crystal Lattices and Unit Cell: Different types of substances are present around us in solids, liquids, and gaseous forms. The solids, liquids, and gases are different in properties from each other due to the arrangement of atoms and molecules. Liquids and gases can flow, whereas solids are rigid, and because of the rigidity possessed by the solids, they have a definite shape and volume. The solids can be classified into two types- crystalline solids and amorphous solids.

What are Crystalline Solids?

Crystalline solids consist of atoms, ions, and molecules arranged in a definite and repeating three-dimensional pattern, containing long-range in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. Examples include snowflakes, diamonds, and table salt.

What is Crystal Lattice and Unit Cell?

The crystal lattice and unit cell are given below:

Crystal Lattice

Crystal Lattice or Space lattice is an arrangement of points regularly repeating in space. As it is a repeating arrangement, to describe the space lattice well, we can choose a small part of the lattice, which produces the complete space lattice when repeated in different directions. This small portion of the lattice is called the ‘unit cell.’

Thus, a regular arrangement of the constituent particles – atoms, ions, or molecules of a crystal in a three-dimensional space is called crystal lattice or space lattice.

Characteristics of a Crystal Lattice

Some of the characteristics of a crystal lattice are as follows:

1. Each point in the crystal lattice represents constituent particles which may be an atom, a molecule, or an ion.
2. Each point in the lattice is called lattice point or lattice site.
3. The points are joined by lines to represent the geometry of the lattice.

Unit Cell

The smallest three-dimensional part of a complete space lattice that repeats in different directions to produce the complete space lattice is called a unit cell.

The unit cell may be considered the fundamental building block of the crystal lattice because the lattice can be constructed by stacking the unit cells.

Parameters of a Unit Cell

A unit cell is characterized by the following parameters which decide the shape and size of the unit cell:

1. It has dimensions \(\rm{a, b, c}\). These edges may or may not be mutually perpendicular.
2. Angles between the edges, i.e., angle \( {\text{α}}\) between the edges \(\rm{b}\) and \(\rm{c}\), angle \( {\text{β}}\) between the edges \(\rm{a}\) and \(\rm{c}\), and angle \( {\text{γ}}\) between the edges \(\rm{a}\) and \(\rm{b}\).

Thus, a unit cell is characterized by six parameters, i.e., axial lengths \(\rm{a, b, c}\), and the axial angles \( {\text{α}}\), \( {\text{β}}\), and \( {\text{γ}}\).

Types of Unit Cell

It was assumed that the particles are present only at the corners of the unit cell. However, it has been observed that the particles may be present not only at the corners but may also be present at some other special positions in addition to those at corners. Hence, the unit cell may broadly be divided into the following categories:

(a) Primitive Unit Cells

The cells in which the constituent particles are present only at the corners are called simple unit cells or primitive unit cells. Thus, there are seven types of primitive unit cells, as given in the table below.

(b) Non-Primitive or Centred Unit Cells

Those unit cells in which the constituent particles are present at the corners of the unit cells and some other positions are called non-primitive unit cells or centred unit cells. There are three types of centred unit cells which are as follows:

i. Face-centred unit cells:  When the particles are present at the corners and at the centre of each face of the unit cell, it is called a face-centred unit cell.

ii. Body-centred unit cell– In addition to the particles at the corners, one particle is present at the centre within the body of the unit cell. It is called a body-centred unit cell.

iii. End-centred unit cell-  When in addition to the particles at the corners, there are particles at the centre of any two opposite faces, it is called end-face centred.

Every crystal lattice does not have all four types of unit cells. Hence, there are only \(14\) types of space lattices corresponding to seven crystal systems.

Arrangement of Unit Cell

There are seven types of unit cells based upon the parameters of the unit cell. These are also called crystal systems or crystal habits because any crystalline solid must belong to any of the unit cells. These different crystal systems, along with their characteristics and examples, are as follows:

System	Axial lengths	Axial Angles	Examples
1. Cubic	\(\rm{a = b = c}\)	\(\alpha  = \beta  = \gamma  = 90^\circ \)	\({\text{Cu}}\), \({\text{NaCl}}\)
2. Tetragonal	\(\rm{a = b} \ne \rm{c}\)	\(\alpha  = \beta  = \gamma  = 90^\circ \)	\({\text{Sn}}\), \({\text{SnO}}_2\)
3. Orthorhombic	\(\rm{a} \ne \rm{b} \ne \rm{c}\)	\(\alpha  = \beta  = \gamma  = 90^\circ \)	\({\text{KnO}}_3\), \({\text{BaSO}}_4\)
4. Monoclinic	\(\rm{a} \ne \rm{b} \ne \rm{c}\)	\(\alpha  = \gamma = 90^\circ \ne \beta\)	\({\text{Na}}_2 {\text{SO}}_4\), \(10{\text{H}}_2 {\text{O}}\)
5. Triclinic	\(\rm{a} \ne \rm{b} \ne \rm{c}\)	\(\alpha  \ne \beta  \ne \gamma  \ne 90^\circ \)	\( {\text{CuS}} { {\text{O}}_4}\), \( 5{\text{H}}_2 { {\text{O}}}\)
6. Rhombohedral	\(\rm{a = b = c}\)	\(\alpha  = \beta  = \gamma  \ne 90^\circ \)	Calcite \(\left(  { {\text{CaC}} { {\text{O}}_ {\text{3}}}}  \right)\), \({\text{HgS}}\) (cinnabar)
7. Hexagonal	\(\rm{a = b} \ne \rm{c}\)	\(\alpha  = \beta  = 90^\circ\), \(\gamma = 120^\circ\)	Graphite, \({\text{ZnO}}\), \({\text{CdS}}\)

Auguste Bravais, a French scientist, found fourteen possible three-dimensional lattices now known as the Bravais Lattice. The following diagram shows these fourteen arrangements.

Calculation of Number of Atoms in a Unit Cell of a Cubic Crystal System

In a crystal lattice, each unit cell touches several other unit cells. Thus, a particle present at the lattice may be shared by a number of unit cells.

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Calculation of Contribution of Atoms Present at Different Lattice Sites

i. An atom at the corner is shared by eight unit cells. Hence, the contribution of each atom present at the corner \( = \frac{1}{8}\).

ii. An atom on the face is shared between two unit cells. Hence, the contribution of each atom on the face \( = \frac{1}{2}\).

iii. An atom present within the body of the unit cell is shared by no other unit cell.

Hence, the contribution of each atom within the body \(= 1\).

iv. An atom present on edge is shared by four unit cells.

Hence, the contribution of each atom on edge \( = \frac{1}{4}\).

Calculation of Number of Atoms Per Unit Cell

1. The number of atoms in a simple (primitive) cubic lattice is that a simple cubic lattice has only eight atoms on the corners. As contribution of each atom is \( = \frac{1}{8}\), therefore, the number of atoms present in the unit cell \( = \frac{1}{8} \times 8 = 1\)
2. The number of atoms in a unit cell of body-centred cubic- This lattice has \(8\) atoms on the corners and one atom within the body.
   Therefore, the contribution of \(8\) atoms present on the corners \( = \frac{1}{8} \times 8 = 1\).
   The contribution of the atom present within the body \(=1\).
   The number of atoms present in the unit cell \( = 1 + 1 = 2\).
3. The number of atoms in a unit cell of the Face-centred cubic- This lattice has \(8\) atoms on the corners and \(6\) atoms on the face (one on each face).
   Contribution by atom on the corners \( = \frac{1}{8} \times 8 = 1\).
   Contribution by atoms on the faces \( = \frac{1}{2} \times 6 = 3\).
   The number of atoms present in the unit cell \(= 1 + 3 = 4\).

Calculation of Space Occupied, i.e., Packing Efficiency

The calculation of the space occupied is explained below:

Packing Efficiency in Simple Cubic Lattice

Suppose the edge length of the unit cell \(= \rm{a}\) and radius of the sphere \(= \rm{r}\).

As the spheres are touching each other, evidently \(\rm{a} = 2\rm{r}\).

Number of spheres per unit cell \( = \frac{1}{8} \times 8 = 1\)

The volume of the sphere \( = \frac{4}{3} {{\pi}} { {\text{r}}^3}\)

The volume of the cube \( {\text{=}} { {\text{a}}^ {\text{3}}} {\text{=}} {\left({ {\text{2r}}}  \right)^ {\text{3}}} {\text{=8}} { {\text{r}}^ {\text{3}}}\)

The packing efficiency \( = \frac{ { {\text{Volume occupied by sphere in unit cell}} \times 100}}{ { {\text{Total volume of the unit cell}}}} {\% }\)

Therefore, the packing fraction \( {\text{=}}\frac{ {\text{4}}}{ {\text{3}}} {{\pi }} { {\text{r}}^ {\text{3}}} {\text{/8}} { {\text{r}}^ {\text{3}}} {\text{=}}\frac{ {{\pi }}}{ {\text{6}}} {\text{=0}} {\text{.524}}\)

The packing efficiency in a simple cubic unit cell \(= 52.4\%\).

Packing Efficiency in Face-centred Cubic Structures

A sphere on the face centre touches the spheres at the corners.

Let the edge length of the cube be a and the radius of each particle be \(\rm{r}\), therefore \(\rm{AC=r+2r+r=4r}\).

From the right-angled triangle \(\rm{ABC}\),

\( {\text{AC}} = \sqrt {\rm{A} {\rm{B}^2} + \rm{B} {\rm{C}^2}} \)

\( = \sqrt { {\rm{a}^2} + {\rm{a}^2}} = \sqrt 2 \rm{a} = 4\rm{r}\) or \( {\text{a=}} \frac{ {\text{4}}}{ {\sqrt {\text{2}} }} {\text{r}}\)

Therefore, the volume of the unit cell \( {\text{=}} { {\text{a}}^ {\text{3}}}\)\( {\left({\frac{ {\text{4}}}{ {\sqrt {\text{2}} }} {\text{r}}} \right)^ {\text{3}}} {\text{=}}\frac{ { {\text{32}}}}{ {\sqrt {\text{2}} }} { {\text{r}}^ {\text{3}}}\)

The packing efficiency \( = \frac{ { {\text{Volume occupied by sphere in unit cell}} \times 100}}{ { {\text{Total volume of the unit cell}}}}\% \)

The packing fraction \( {\text{=}}\frac{ { {{16 \pi }} { {\text{r}}^ {\text{3}}} {\text{/3}}}}{{ {\text{32}} { {\text{r}}^ {\text{3}}} {\text{/}}\sqrt {\text{2}} }} {\text{=}}\frac{ { {{\pi }}\sqrt {\text{2}} }}{ {\text{6}}} {\text{=0}} {\text{.74}}\)

The packing efficiency in a Face-centred cubic unit cell \(= 74\%\).

Packing Efficiency in Body-Centred Cubic Structure

As the sphere at the body centre touches the spheres at the corners, the body diagonal, \(\rm{AD=4r}.\)

Further, face diagonal, \( {\text{AC=}}\sqrt { {\text{A}} { {\text{B}}^ {\text{2}}} {\text{ + B}} { {\text{C}}^ {\text{2}}}} \)

\( {\text{=}}\sqrt { { {\text{a}}^ {\text{2}}} {\text{ + }} { {\text{a}}^ {\text{2}}}} {\text{=}}\sqrt {\text{2}} {\text{a}}\)

And body diagonal, \( {\text{AD=}}\sqrt { {\text{A}} { {\text{C}}^ {\text{2}}} {\text{ + C}} { {\text{D}}^ {\text{2}}}}\)

\( {\text{=}}\sqrt { {\text{2}} { {\text{a}}^ {\text{2}}} {\text{ + }} { {\text{a}}^ {\text{2}}}} {\text{=}}\sqrt  {\text{3}} {\text{a=4r}}\) or \( {\text{a=}}\frac{ {\text{4}}}{ {\sqrt {\text{3}} }} {\text{r}}\)

Therefore, the volume of two sphere \( {{=2 \times }}\frac{ {\text{4}}}{ {\text{3}}} {{\pi }} { {\text{r}}^ {\text{3}}} {\text{=}}\frac{ { {\text{64}} { {\text{r}}^ {\text{3}}}}}{ { {\text{3}}\sqrt {\text{3}} }}\)

Number of spheres per unit cell \( = 8 \times \frac{1}{8} + 1 = 2\)

The volume of \(2\) spheres \( {{=2 \times }}\frac{ {\text{4}}}{ {\text{3}}} {{\pi }} { {\text{r}}^ {\text{3}}} {\text{=}}\frac{ {\text{8}}}{ {\text{3}}} {{\pi }} { {\text{r}}^ {\text{3}}}\)

The packing efficiency \( = \frac{ { {\text{Volume occupied by sphere in unit cell}} \times 100}}{ { {\text{Total volume of the unit cell}}}} {{\% }}\)

Packing fraction \( {\text{=}}\frac{ {\frac{ {\text{8}}}{ {\text{3}}} {{\pi }} { {\text{r}}^ {\text{3}}}}}{ {\frac{ { {\text{64}} { {\text{r}}^ {\text{3}}}}}{ { {\text{3}}\sqrt {\text{3}} }}}} {\text{=}}\frac{ { {{\pi }}\sqrt {\text{3}} }}{ {\text{8}}} {\text{=0}} {\text{.68}}\)

The packing efficiency in a Body-centred cubic unit cell \(= 68\%\)

From the above discussion, crystal lattice and unit cell differences can be easily understood. The packing efficiency of various lattices can be used in solving the examples.

Solved Examples on Crystal Lattices and Unit Cell

Q.1. An ionic compound made of atoms \(\rm{A}\) and \(\rm{B}\) has a face-centred cubic arrangement in which atoms \(\rm{A}\) are at the corners and atoms \(\rm{B}\) are at the face centres. If one of the atoms is missing from the corner, what is the simplest formula of the compound?
Ans: The number of atoms of \(\rm{A}\) at the corners \(= 7\)
Therefore, contribution atoms of \(\rm{A}\) towards unit cell \( = 7 \times \frac{1}{8} = \frac{7}{8}\)
The number of atoms \(\rm{B}\) at face-centred \(= 6\).
Contribution of atom B towards unit cell \( = 6 \times \frac{1}{2} = 3\)
The ratio of \( {\text{A:B=}}\frac{ {\text{7}}}{ {\text{8}}} {\text{:3=7:24}}\)
Therefore, the formula is \(\rm{A}_7 \rm{B}_{24}.\)

Q.2. Calculate the number of unit cells in \(8.1\,\rm{g}\) of aluminium if it crystallizes in a face-centred cubic structure.(Atomic mass of \(\rm{Al}= 27\,\rm{g}\,\rm{mol}^{-1})\).
Ans: \(1\,\rm{mole}\) of \(\rm{Al} = 27\,\rm{g} = 6.022 \times 10^{23}\) atoms
The number of \(\rm{Al}\) atoms present in \(8.1\,\rm{g}\) of \( {\text{Al}} = \frac{ {6.022 \times { {10}^ {23}}}}{ {27}} \times 8.1 = 1.806 \times {10^ {23}}\)
As face-centred cubic unit cell contains \(4\) atoms, therefore,
The number of unit cells present \( = \frac{ {1.806 \times { {10}^ {23}}}}{4} = 4.515 \times {10^ {22}}.\)

Summary

In this article, we understood the crystal lattice and unit cell definition and focussed on the unit cell types and calculations used. We also studied the seven different types of unit cells available and calculated the number of atoms in the unit cell and the space occupied, i.e., Packing Efficiency.

FAQs on Crystal Lattices and Unit Cell

Q1. What are the seven types of crystals?
Ans: The seven types of crystals are as follows:

Q.2. What are the crystal lattice and unit cells?
Ans: A regular arrangement of the constituent particles- atoms, ions, or molecules of a crystal in a three-dimensional space is called crystal lattice or space lattice.
The unit cell is the smallest three-dimensional portion of a complete space lattice that repeatedly, in different directions, produces the complete space lattice.
The unit cell may be considered the fundamental building block of the crystal lattice because the lattice can be constructed by stacking the unit cells.

Q3. What is the difference between crystal lattice and lattice point?
Ans: A regular arrangement of the constituent particles- atoms, ions, or molecules of a crystal in a three-dimensional space is called crystal lattice or space lattice whereas the lattice is made up of many points, and these represent one part of the crystal and therefore is known as lattice point in the crystal lattice.

Q.4. What is the difference between a unit cell and a single crystal?
Ans: The unit cell is the fundamental building block of the crystal lattice because the lattice can be constructed by stacking the unit cells, whereas a single crystal can have many unit cells. 

Q.5. How are the unit cell and crystal lattice related?
Ans: Space lattice is a regularly repeating arrangement of points in space. As it is a repeating arrangement, to describe the space lattice completely, we can choose a small part of the lattice, which produces the complete space lattice when repeated in different directions. This small portion of the lattice is called the ‘unit cell’.