Solid State - Result Question 6

####6. The radius of the largest sphere which fits properly at the centre of the edge of a body centred cubic unit cell is (Edge length is represented by ’ $a$ ‘) (2019 Main, 11 Jan II)

(a) $0.134 a$

(b) $0.027 a$

(c) $0.047 a$

(d) $0.067 a$

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Answer:

Correct Answer: 6. (d)

Solution:

  1. For body centred cubic bcc structure,

radius $(R)=\frac{\sqrt{3}}{4} a$

Where, $a=$ edge length

According to question, the structure of cubic unit cell can be shown as follows:

$\therefore \quad a=2(R+r)$

On substituting the value of $R$ from Eq. (i) to Eq. (ii), we get

$$ \frac{a}{2}=\frac{\sqrt{3}}{4} a+r $$

$$ \begin{aligned} & r=\frac{a}{2}-\frac{\sqrt{3}}{4} a=\frac{2 a-\sqrt{3} a}{4} \ & r=\frac{a(2-\sqrt{3})}{4} \ & r=0.067 a \end{aligned} $$



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