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Salt AB has a zinc blende structure. The radius of ${A^{2 + }}$and ${B^{2 - }}$ions are $0.7$ $\overset{​o}{​A}\,$ and $1.8$ $\overset{​o}{​A}\,$ respectively. The edge length of AB unit cell is :(A) $2.5$ $\overset{​o}{​A}\,$(B) $5.09$ $\overset{​o}{​A}\,$(C) $5$ $\overset{​o}{​A}\,$(D) $5.77$ $\overset{​o}{​A}\,$

Last updated date: 07th Aug 2024
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Hint: The edge length can be defined as the length of the side of the cubic unit cell which is made up of atoms of constituent elements. The edge length of the Zinc blende unit cell can be found out by -
$r_A^ + + r_B^ - = \dfrac{{a\sqrt 3 }}{4}$
Where $r_A^ +$ is the radius of cation
$r_B^ -$ is the radius of the anion
And ‘a’ is the edge length of the unit cell.

Complete step by step solution:
Given :
Salt AB has structure = zinc blende structure
Radius of ${A^{2 + }}$ion = $0.7$ $\overset{​o}{​A}\,$
Radius of ${B^{2 - }}$ion = $1.8$ $\overset{​o}{​A}\,$
To find :
Edge length of AB unit cell
As the structure is a zinc blende structure. We know the zinc blende structure is made of Zinc atoms and Sulphur atoms. The structure consists of a cubic lattice in which Sulphur atoms occupy corners and face centre sites while the Zinc atoms occupy half of the tetrahedral sites.
Further, we know that for zinc blende structure, the relation between radius and edge length is given by -
$r_A^ + + r_B^ - = \dfrac{{a\sqrt 3 }}{4}$
Where $r_A^ +$ is the radius of cation
$r_B^ -$ is the radius of anion
And ‘a’ is the edge length of a unit cell.
By putting the values in the formula, we can find the edge length as -
$0.7 + 1.8 = \dfrac{{a\sqrt 3 }}{4}$
$\dfrac{{a\sqrt 3 }}{4}$ = $2.5$
$a\sqrt 3$ = 10
And $a$ = $\dfrac{{10}}{{1.732}}$
‘a’ = $5.77$ $\overset{​o}{​A}\,$

So, the option (D) is the correct option.

Note: It must be noted that Zinc blende structure has atoms at the face centre of the unit cell also. So, it is different from a simple unit cell. The edge length of a simple unit cell can find out as- ‘a’ = $r_A^ + + r_B^ -$
Similarly, the edge length of different types of unit cells is different. It should not be mixed.