Question

The edge length of the unit cell of a bcc lattice of a metal is $350\text{pm}$
What is the length of the body diogonal?
[1 Mark]

• A. $5.92\stackrel{o}{A}$
• B. $3.98\stackrel{o}{A}$
• C. $2.96\stackrel{o}{A}$
• D. $8.8\stackrel{o}{A}$

Answer 1

The correct option is A $5.92\stackrel{o}{A}$

In a bcc structure, the atoms toch each other along the body diagonal.

let the radius of the atom $=r$
Given, edge length of unit cell = $a$
From the diagram,
$A{B}^2=B{C}^2+A{C}^2$
$B{C}^2=B{D}^2+C{D}^2=2a^2$
$\therefore A{B}^2=2a^2+a^2=3a^2$
$\therefore AB=\sqrt{3}a$
But $AB=4r$
$\therefore 4r=\sqrt{3}a\Rightarrow r=\sqrt{3}a/4$
$\Rightarrow r=\sqrt{3}\times \frac{350}{4}\text{pm}$
$=\sqrt{3}\times \frac{350}{4}\times {10}^{-12}\text{m}$
$=148.75\times {10}^{-10}\text{cm}$
$=1.48\stackrel{o}{A}$

So length of the body diogonal = $4r=4\times 1.48\stackrel{o}{A}=5.92\stackrel{o}{A}$