# GATE MT 2016 Q13. For an ideal hexagonal-closed packed structure, the c/a ratio and packing efficiency respectively are

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Askmemetallurgy Official

Asked: 2020-04-08T04:00:19+05:30
2020-04-08T04:00:19+05:30In: Physical Metallurgy & Heat treatment

12.5%(A) 1.633 and 52% ( 1 voter )
87.5%(B) 1.633 and 74% ( 7 voters )
0%(C) 1.733 and 68%
0%(D) 1.733 and 74%

Based On 8 Votes

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Correct answer is option B. For an ideal hexagonal-closed packed structure, c/a= 2√2/3= 1.633 and packing efficiency= 0.74 = 74% Here the unit cell (equivalent to 3 primitive unit cells) is a hexagonal prism containing six atoms (if the particles in the crystal are atoms). Indeed, three are the atomRead more

Correct answer is option B.

For an ideal hexagonal-closed packed structure,

c/a= 2√2/3= 1.633 and packing efficiency= 0.74 = 74%

Here the unit cell (equivalent to 3 primitive unit cells) is a hexagonal prism containing six atoms (if the particles in the crystal are atoms). Indeed, three are the atoms in the middle layer (inside the prism); in addition, for the top and bottom layers (on the bases of the prism), the central atom is shared with the adjacent cell, and each of the six atoms at the vertices is shared with other five adjacent cells. So the total number of atoms in the cell is 3 + (1/2)×2 + (1/6)×6×2 = 6.

Each atom touches other twelve atoms. let ‘a’ be the side length of the base of the prism and ‘c’ be its height. The latter is twice the distance between adjacent layers,

i. e., twice the height of the regular tetrahedron whose vertices are occupied by (say) the central atom of the lower layer, two adjacent non-central atoms of the same layer, and one atom of the middle layer “resting” on the previous three. The edge of this tetrahedron is a. If a=2r , then its height can be easily calculated to be , .It is then possible to calculate the APF as follows:

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