Question

Calculate the efficiency of packing in case of a metal crystal for the following crystal structure (with the assumptions that atoms are touching each other):
(i) Simple cubic
(ii) Body-centered cubic
(iii) Face-centered cubic

Answer 1

In crystallography, crystal structure is an illustration of the arrangement of atoms, ions or molecules in a crystalline material. The lengths of the principal axes, or edges, of the unit cell and the angles between them are the lattice constants, called as cell parameters or lattice parameters.

Complete step by step answer:
i.SIMPLE CUBIC:
The efficiency of packing of simple cubic unit cell is given below:
A simple cubic cell contains one atom per unit cell.
In which $a = 2r$, where $a$ is the edge length and $r$ is the radius of the atom.
The total volume of the unit cell =${a^3} = 8{r^3}$.
Therefore,
The packing efficiency is given by =Total volume of unit cell Volume of one sphere​×100
Packing efficiency = $\dfrac{{\dfrac{4}{3}\pi {r^3}}}{{8{r^3}}} \times 100 = 52.4\%$

ii.BODY-CENTERED CUBIC:
The efficiency of packing of body-centred unit cell is given below:
A body-centred cubic cell contains two atoms per unit cell.
In which $3a = 4r$, where $a$ is the edge length and $r$ is the radius of the atom.
The total volume of the unit cell =${a^3} = \dfrac{{64}}{{3\sqrt 3 }}{r^3}$.
The packing efficiency is given by =Total volume of unit cell Volume of two spheres​×100
Packing efficiency =$\dfrac{{\dfrac{8}{3}\pi {r^3}}}{{\dfrac{{64}}{{3\sqrt 3 }}{r^3}}} \times 100 = 68\%$

iii.FACE-CENTERED CUBIC:
The efficiency of packing in case of face-centered cubic cell (with the assumptions that atoms are touching each other) is given below:
A face-centered cubic cell contains four atoms per unit cell.
In which $a = 2\sqrt 2 r$, where $a$ is the edge length and $r$ is the radius of the atom.
The total volume of the unit cell =${a^3} = 16\sqrt 2 {r^3}$.
The packing efficiency is given by =Total volume of unit cell Volume of four spheres​×100
Packing efficiency =$\dfrac{{\dfrac{{16}}{3}\pi {r^3}}}{{16\sqrt 2 {r^3}}} \times 100 = 74\%$

Note:
There are 32 different combinations of elements of symmetry of a crystal which are called 32 systems. Among all 7 types of basic or primitive unit cells have been recognized among crystals. They are cubic, orthorhombic, hexagonal, tetragonal, monoclinic, triclinic and rhombohedral.