Question

The vacant space in bcc lattice unit cell is:

• A. 0.23
• B. 0.32
• C. 0.26
• D. 0.48

Hint:

BCC unit cell includes eight atoms at the corner of the cube and one atom at the center of the unit cell. We can find % vacant space by following the equation.
% Vacant space = 100 - % Volume occupied by atoms in a unit cell

Answer 1

The Correct Option is B

 In a body-centered cubic (bcc) lattice, the arrangement of atoms results in some space within the unit cell that is not occupied by atoms. This unoccupied space is referred to as the "vacant space" or "void space." The calculation of vacant space in a bcc lattice can be determined through the geometry of the lattice.

• The bcc unit cell contains 1 atom at the center and 8 atoms at the corners, each contributing 1/8th of an atom to the unit cell, making a net total of 2 atoms per unit cell.
• The relationship between the lattice parameter \(a\) (edge length) and the atomic radius \(r\) in a bcc lattice is given by: \(a = \frac{4r}{\sqrt{3}}\). This is because the atoms touch along the body diagonal.
• The volume of the unit cell is computed as \(a^3\).
• The volume occupied by the atoms in the bcc unit cell can be calculated using the volume of a sphere: \(V_{\text{atom}} = 2 \times \frac{4}{3}\pi r^3\).
• The packing efficiency (fraction of volume occupied by atoms in the unit cell) of a bcc structure is known to be approximately 68%. Therefore, the vacant space, in percentage, is: \(100\% - 68\% = 32\%\).

Thus, the vacant space in a bcc lattice unit cell is \(0.32\), making

0.32

the correct answer.