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Crystal Lattices and Unit Cells 

What is a Crystal?

A crystalline solid consist of a large number of small units, called crystals, each of which possesses a definite geometric shape bounded by plane faces. The crystals of a given substance produced under a definite set of conditions are always of the same shape

All crystals consists of regularly repeating array of atoms, molecules or ions which are the structural units (or basic units). It is much more convenient to represent each unit of pattern by a point, called lattice point, rather than drawing the entire unit of pattern. This results in a three dimensional orderly arrangement of points called a space lattice or a crystal lattice

Thus, a space lattice or crystal lattice  may be defined as a regular three dimensional arrangement of identical points in space or it can be defined as an array of points showing how molecules, atoms or ions are arranged at different sites in three dimensional space.

It must be noted that

  • Each lattice point has the same environment as that of any other point in the lattice

  • The constituent particles have always to be represented by a lattice point, irrespective of whether it contains a single atom or more than one atoms

What is a Unit Cell ? 

A unit cell is the smallest repeating unit in space lattice which when repeated over and over again results in a crystal of the given substance. Unit cell may also be  defined as a three dimensional group of lattice points that generates the whole lattice on repetition.

A unit cell is characterised by:

  • its dimensions along the three edges, a, b and c. 

  • angles between the edges, α (between b and c) β (between a and c) and γ (between a and b). Thus, a unit cell is characterised by six parameters, a, b, c, α, β and γ.

Types of unit cells

  • Simple unit cell or Primitive unit cell -  A unit cell having lattice points only at the corners is called simple, primitive or basic unit cell. A crystal lattice having primitive unit cell is called simple crystal lattice

  • Face centred cubic lattice (fcc) – A unit cell in which the lattice point is at the centre of each face as well as at the corner.

  • Body centred cubic lattice (bcc) –  A unit cell having a lattice point at the centre of the body as well as at the corners. Another type of unit cell, called end – centred unit cell is possible for orthorhombic and monoclinic crystal types. In an end centred there are lattice points in the face centres of only one set of faces in addition to the lattice pints at the corners of the unit cell

The various types of unit cells possible for different crystal classes (in all seven) are given below in tabular form

Crystals class

Axial distances

Angles

Possible types of unit cells

Examples

Cubic

a = b = c

α = β = γ = 90°

Primitive, Body centred face centred

Copper , KCl, NaCl zinc blende, diamond

Tetragonal

a = b ≠ c

α = β = γ = 90°

Primitive, body centred

SnO2, White tin, TiO2

Orthorhombic

a ≠ b ≠ c

α = β = γ = 90°

Primitive body centred, face centred end centred

Rhombic sulphur, KNO3, CaCO3

Hexagonal

a = b ≠ c

α = β = 90°

γ = 120°

Primitive

Graphite, Mg, ZnO

Trigonal or Rhombohedral

a = b = c

α = β = γ ≠90°

Primitive

(CaCO3) Calcite, HgS(Cinnabar)

Monoclinic

a ≠ b ≠ c

α = β = 90°

γ ≠ 90°

Primitive and end centred

Monoclinic sulphur, Na2SO4.10H2O

Triclinic

a ≠ b ≠ c

α ≠ β ≠ γ ≠90°

Primitive

K2Cr2O7, CuSO4.5H2O

Bravais Lattice

The French crystallographer August Bravais in 1848 showed from geometrical consideration that there can be only 14 different ways in which similar points can be arranged in a three dimensional space. 

Thus the total no. of space lattices belonging to all the seven basic crystal system but together is only 14.

The Bravais space lattices associated with various crystal system are show in fig below

 

Calculation of number of particles per unit cell

The no of atom in a unit cell can be calculated by keeping in view following points

  • An atom at the corners is shared by eight unit cells. Hence the contribution of an atom at the corner to a particular cell = 1/8.

  • An atom at the face is shared by two unit cells. Hence the contribution of an atom at the face to a particular cell = 1/2

  • An atom at the edge centre is shared by four unit cells in the lattice and hence contributes only 1/4 to a particular unit cell.

  • An atom at the body centre of a unit cell belongs entirely to it, so its contribution = 1

Simple Cubic Lattice 

Body Centred Cubic (BCC)

Face Centred Cubic (FCC)

There are eight atoms at the corners. Each corner atom makes 1/8 contribution to the unit cell.

∴ No. of atoms present in the unit cell = 1/8 × 8 = 1

BCC has 8 atoms at the corners and one atom, within the body. Each corner atom makes 1/8 contribution and the contribution of atom within the body = 1

∴ No of atoms present in bcc = 1/8 × 8 (at corner) + 1(at the body centre)= 1+1 =2

Fcc has 8 atoms at the corners and 6 atoms on the faces (one on each face)

Contribution by atoms at the corners = 1/8 × = 1

Contribution by atom on the face = 1/2 × 6 = 3

∴ Number of atoms present in fcc unit cell = 1+3 = 4  

The number of atoms per unit cell is in the same ratio as the stoichiometry of the compound. Hence it helps to predict the formula of the compound

Solved Examples

Example1. 

A compound formed by elements A and B has a cubic structure in which A atoms are at the corners of the cube and B atoms are at face centres. Derive the formula of the compound.

Solution:         

As ‘A’ atom are present at the 8 corners of the cube therefore no of atoms of A in the unit cell = 1/8 × 8 = 1 

As B atoms present at the face centres of the cube, therefore no of atoms of B in the unit cell = 1/2 × 6 = 3 

Hence the formula of compound is AB3

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Example 2. 

Potassium crystallizes in a body centred cubic lattice. What is the approximate number of until cells in 4.0 g of potassium? Atomic mass of potassium = 39.

Solution:         

In bcc unit cell there are 8 atoms at the corners of the cube and one atom at the body centre 

 ∴ No of atoms per unit = 8 × 1/8 + 1 = 2 

 No of atoms is 4.0 g of potassium = 4/39 × 6.023 ×1023 

∴ No of unit cells in 4.0 g potassium = 4/39 × 6.023 × 1023 / 2 = 3.09 ×1022 

Question 1: A unit cell in which the lattice point is at the centre of each face as well as at the corner is called

a. simple unit cell

b. primitive unit cell.

c. body centred unit cell.

d. face centred unit cell.

Question 2: Which of the following crystall classes is reprented by α = β = γ = 90° ?

a. Tetragonal

b. Trigonal

c. Triclinic

d. Rhombohedral

Question 3: For hexagonal cubic system

a. a = b = c

b. a = b ≠ c

c. a ≠ b = c

d. a ≠ b ≠ c

Question 4: Number of particles present in one BBC system is 

a. 1

b. 2

c. 3

d. 4

Question 5: Total number of Bravais Lattice is

a. 3

b. 7

c. 14

d. 16

Q.1

Q.2

Q.3

Q.4

Q.5

d

a

b

b

c

Related Resources:- 

To read more, Buy study materials of Solid State comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Chemistry here.

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