In which of the following close-packed structures, the radii r of their spheres respectively is correctly represented?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\text{ccp, simple cubic : }\frac{a}{2\sqrt{2}}, \frac{a}{2}\)
The radius r of the spheres in a cubic unit cell of edge a follows from the direction in which the spheres touch:
Simple cubic: spheres touch along the edge: \( a = 2r \Rightarrow r = \frac{a}{2} \)
Body centred cubic: spheres touch along the body diagonal: \( 4r = \sqrt{3}\,a \Rightarrow r = \frac{\sqrt{3}}{4}a \)
Face centred cubic (ccp): spheres touch along the face diagonal: \( 4r = \sqrt{2}\,a \Rightarrow r = \frac{a}{2\sqrt{2}} \)
Why the other options are incorrect
Option (1) — Incorrect. √3a/4 is the bcc value, not the hcp value, and a/(2√2) is the ccp value, not the bcc value.
Option (2) — Incorrect. The ccp value is right, but in the hexagonal (hcp) cell the spheres touch along the edge of the base, so r = a/2 there.
Option (3) — Incorrect. Neither value fits: bcc needs √3a/4 and simple cubic needs a/2 (2a is larger than the cell edge).
Note: This question is from the pre-rationalised NCERT book (2022-23). The topic is not there in the latest NCERT book (2026-27), and it is also out of the CUET 2026 syllabus.
NCERT (Pre-rationalised edition, 2022-23):
Packing Efficiency in hcp and ccp Structures
