Question

Which of the following statement(s) is/are true regarding triangular voids?

• A. Triangular voids are formed in 1-D close packing
• B. Coordination number of triangular void is 3
• C. $\text{Maximum radius of triangular void}=0.155\times \text{radius of sphere}$
• D. $\text{Maximum radius of triangular void}=0.225\times \text{radius of sphere}$

Answer 1

Answer: (B), (C)

$\text{Maximum radius of triangular void}=0.155\times \text{radius of sphere}$

Triangular voids are found in 2-D planar close packed structure where three spheres are in contact with each other. Thus, the coordination number of triangular void is 3.


Here,
R is the radius of the sphere
r is the radius of triangular void

In the triangular void, the triangle formed by joining the centres of the spheres gives a equilateral triangle ABC. Thus, the angle of triangle, $\angle ABC$ is ${60}^{\circ }$
Hence, the angle of triangle, $\angle OBC$ will be ${30}^{\circ }$
$\therefore$
In $△BOD$
$Cos{30}^{\circ }=\frac{BD}{BO}$

$Cos{30}^{\circ }=\frac{R}{R+r}$

$\frac{1}{Cos{30}^{\circ }}=\frac{R+r}{R}$

​​​​​​​$\frac{r}{R}=0.155$
$r=0.155\times R$
$\text{Maximum radius of triangular void}=0.155\times \text{radius of sphere}$

Thus, statements (b) and (c) are correct.