Answer:
Explanation:
The percentage of unit cell volume = Volume of atoms/Volume of unit cell
Volume of sphere = [tex]\frac{4 }{3} \pi r^2[/tex]
a) Percentage of unit cell volume occupied by atoms in face- centered cubic lattice:
let the side of each cube = a
Volume of unit cell = Volume of cube = a³
Radius of atoms = [tex]\frac{a\sqrt{2} }{4}[/tex]
Volume of each atom = [tex]\frac{4 }{3} \pi (\frac{a\sqrt{2}}{4})^3[/tex] = [tex]\frac{\pi *a^3\sqrt{2}}{24}[/tex]
Number of atoms/unit cell = 4
Total volume of the atoms = [tex]4 X \frac{\pi *a^3\sqrt{2}}{24} = \frac{\pi *a^3\sqrt{2}}{6}[/tex]
The percentage of unit cell volume = [tex]\frac{\frac{\pi *a^3\sqrt{2}}{6}}{a^3} =\frac{\pi *a^3\sqrt{2}}{6a^3} = \frac{\pi \sqrt{2}}{6}[/tex] = 0.7405
= 0.7405 X 100% = 74.05%
b) Percentage of unit cell volume occupied by atoms in a body-centered cubic lattice
Radius of atoms = [tex]\frac{a\sqrt{3} }{4}[/tex]
Volume of each atom =[tex]\frac{4 }{3} \pi (\frac{a\sqrt{3}}{4})^3[/tex] =[tex]\frac{\pi *a^3\sqrt{3}}{16}[/tex]
Number of atoms/unit cell = 2
Total volume of the atoms = [tex]2X \frac{\pi *a^3\sqrt{3}}{16} = \frac{\pi *a^3\sqrt{3}}{8}[/tex]
The percentage of unit cell volume = [tex]\frac{\frac{\pi *a^3\sqrt{3}}{8}}{a^3} =\frac{\pi *a^3\sqrt{3}}{8a^3} = \frac{\pi \sqrt{3}}{8}[/tex] = 0.6803
= 0.6803 X 100% = 68.03%
c) Percentage of unit cell volume occupied by atoms in a diamond lattice
Radius of atoms = [tex]\frac{a\sqrt{3} }{8}[/tex]
Volume of each atom = [tex]\frac{4 }{3} \pi (\frac{a\sqrt{3}}{8})^3[/tex] = [tex]\frac{\pi *a^3\sqrt{3}}{128}[/tex]
Number of atoms/unit cell = 8
Total volume of the atoms = [tex]8X \frac{\pi *a^3\sqrt{3}}{128} = \frac{\pi *a^3\sqrt{3}}{16}[/tex]
The percentage of unit cell volume = [tex]\frac{\frac{\pi *a^3\sqrt{3}}{16}}{a^3} =\frac{\pi *a^3\sqrt{3}}{16a^3} = \frac{\pi \sqrt{3}}{16}[/tex] = 0.3401
= 0.3401 X 100% = 34.01%