Crystal Lattices and Unit Cells MCQ - Practice Questions & Answers

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Crystal Lattices and Unit Cells

Crystal Lattices and Unit Cells
A portion of the three-dimensional crystal lattice and it's unit cell as shown in Fig. given below:

In the three-dimensional crystal structure, a unit cell is characterised 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 characterised by six parameters a, b, c, α, β and γ.
These parameters of a typical unit cell are shown in Fig. given below:

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.

Centred 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 centred unit cell. Centred unit cells are of three types:

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

Crystal system	Bravias lattices	Intercepts	Interfacial angle	Examples
Cubic	Primitive, face-centred, body-centred = 3	a = b = c	⍺ = β = 𝛄 = 90℃	Ag, Au, Hg, Pb, diamond, NaCl, ZnS
Orthorhombic	Primitive, face centered, body centered, end centered = 4	a ≠ b ≠ c	⍺ = β = 𝛄 = 90℃	K₂SO₄,KNO₂, BaSO₄, Rhombic Sulphur
Tetragonal	Primitive, body-centred = 2	a = b ≠ c	⍺ = β = 𝛄 = 90℃	TiO₂, SnO₂, CaSO₄, White Tin
Monoclinic	Primitive, end centered = 2	a ≠ b ≠ c	⍺ = β = 𝛄 ≠ 90℃	CaSO₄.2H₂O
Triclinic	Primitive = 1	a ≠ b ≠ c	⍺ ≠ β ≠ 𝛄 ≠ 90℃	CuSO₄.5H₂O, K₂Cr₂O₇, H₃BO₃
Hexagonal	Primitive = 1	a = b ≠ c	⍺ = β = 90℃ 𝛄 = 120℃	Zn, Mg, Cd, SiO₂, Graphite, ZnO
Rhombohedral	Primitive = 1	a = b = c	⍺ = β = 𝛄 ≠ 90℃	Bi, As, Sb, CaCO₃, HgS
Total = 14

Unit Cells of 14 Types of Bravais Lattices

Number of Atoms in a Unit Cell

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
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 of 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 centre 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 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 cells 8x(1/8) = 1 atom.

BodyCentred 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 centre. 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

FaceCentred Cubic Unit Cell
A face-centred cubic (fcc) unit cell contains atoms at all the corners and at the centre 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) open structure (b) 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