Crystal Lattices and Unit Cells - Difference, Parameters, Properties, FAQs

The crystal lattice and the unit cell are the three-dimensional, repeating patterns of the atomic, ionic, or molecular positions of atoms in a crystal. Such a structured pattern extends and gives an orderly structure to the material. The Unit Cell is the smallest building block of the crystal lattice that can be repeated in space to form the entire lattice.

In the article, we cover the topic of crystal lattice which is the sub-topic of the chapter on Solid states. it is important for board exams JEE Mains Exam, NEET Exam, and other entrance exams.

Crystal Lattices and Unit Cells

A portion of the three-dimensional crystal lattice and its unit cell as shown in Fig below:

In the three-dimensional crystal structure, a unit cell is characterized by:
(i) its dimensions along the three edges a, b, and c. These edges may or may not be mutually perpendicular.
(ii) 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 characterized by six parameters a, b, c, α, β, and γ.
These parameters of a typical unit cell are shown in Fig given below:

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Primitive and Centred Unit Cells

Primitive Unit Cells
When constituent particles are present only on the corner positions of a unit Cell, it is called as primitive unit cell.

Centered Unit Cells
When a unit cell contains one or more constituent particles present at positions other than corners in addition to those at corners, it is called a centered unit cell. Centered unit cells are of three types:

• Body-centered Unit Cells: Such a unit cell contains one constituent particle (atom, molecule, or ion) at its body center besides the ones that are at its corners.
• Face-centered Unit Cells: Such a unit cell contains one constituent particle present at the center of each face, besides the ones that are at its corners.
• End-centered Unit Cells: In such a unit cell, one constituent particle is present at the center of any two opposite faces besides the ones present at its corners.

Inspection of a wide variety of crystals leads to the conclusion that all can be regarded as conforming to one of the seven regular figures. These basic regular figures are called seven-crystal systems.

Seven Primitive Unit Cells and Their Possible Variations as Centred Unit Cells

Unit Cells of 14 Types of Bravais Lattices

Any crystal lattice is made up of a very large number of unit cells and every lattice point is occupied by one constituent particle (atom, molecule or ion).

Primitive Cubic Unit Cell

The primitive cubic unit cell has atoms only at its corner. Each atom at a corner is shared between eight adjacent unit cells as shown in Fig. given below:


four unit cells in the same layer and four unit cells in the upper (or lower) layer. Therefore, only 1/8th of an atom (or molecule or ion) actually belongs to a particular unit cell. In Fig. given below, a primitive cubic unit cell has been depicted in three different ways. Each small sphere in this figure represents only the center of the particle occupying that position and not its actual size. Such structures are called open structures.

The arrangement of particles is easier to follow in open structures as shown in the figure given below depicts a space-filling representation of the unit cell with actual particle size

The figure given below shows the actual portions of different atoms present in a cubic unit cell. In all, since each cubic unit cell has 8 atoms on its corners, the total number of atoms in one unit cell 8x(1/8) = 1 atom.


Body Centred Cubic Unit Cell

A body-centred cubic (bcc) unit cell has an atom at each of its corners and also one atom at its body center. The figure given below depicts (a) open structure (b) space-filling model and (c) the unit cell with portions of atoms actually belonging to it. It can be seen that the atom at the body centre wholly belongs to the unit cell in which it is present.

Thus in a body-centered cubic (bcc) unit cell:

• 8 corners x 1/8 per corner atom = 8 x 1/8 = 1 atom
• 1 body centre atom = 1 x 1 = 1 atom
  Thus, total number of atoms per unit cell = 2 atoms

Face Centred Cubic Unit Cell

A face-centered cubic (fcc) unit cell contains atoms at all the corners and at the center of all the faces of the cube. It can be seen in the figure given below, that each atom located at the face-centre is shared between two adjacent unit cells and only ½ of each atom belongs to a unit cell.

The fig. given below depicts (a) an open structure (b) a space-filling model and (c) the unit cell with portions of atoms actually belonging to it.

Thus, in a face-centred cubic (fcc) unit cell:

• 8 corners atoms x 1/8 atom per unit cell = 8 x 1/8 = 1 atom
• 6 face-centred atoms x 1/2 atom per unit cell = 6 x 1/2 = 3 atoms
  Thus, total number of atoms per unit cell = 4 atoms

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Some Solved Examples

Example 1
Question: The smallest repeating pattern which when repeated in 3-D results in the crystal of substance is called:
1) Space lattice
2) Crystal lattice
3) (correct) Unit cell
4) Bravais lattice

Solution: The unit cell is the smallest repeating unit in the crystal which has all the properties of a crystal. Hence, the answer is the option (3).

Example 2
Question: The most unsymmetrical crystal system is:
1) Cubic
2) Hexagonal
3) (correct) Triclinic
4) Orthorhombic

Solution: The triclinic crystal system has the parameters a n_eq b n_eq c and alpha n_eq beta n_eq gamma n_eq90⁰ . Hence, the answer is the option (3).

Example 3
Question: The crystal system of a compound with unit cell dimensions a=0.387,nm,b=0.387,nm,c=0.504,nmandgamma=120⁰ is:
1) (correct) Hexagonal
2) Cubic
3) Rhombohedral
4) Orthorhombic

Solution: For hexagonal systems, the conditions are a = b neq c and alpha=beta=90⁰,gamma=120⁰ . Thus, the answer is the option (1).

Example 4
Question: For which of the given crystal families does the following relation hold? a not equal b not equal c and alpha=gamma=90⁰,betaneq90⁰
1) (correct) Monoclinic
2) Triclinic
3) Orthorhombic
4) Hexagonal

Solution: The given relation corresponds to the monoclinic crystal system. Hence, the answer is the option (1).

Example 5
Question: The crystal system characterized by a = b = c and alpha=beta=gamma=90⁰ is:
1) Cubic
2) Tetragonal
3) (correct) Orthorhombic
4) Rhombohedral

Solution: The cubic crystal system has equal edge lengths and all angles equal to 90⁰. Therefore, the answer is the option (1).

NCERT Chemistry Notes:

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