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The volume of the HCP unit cell is...

The volume of the HCP unit cell is

A

`24sqrt(2)r^(3)`

B

`(16)/sqrt(2)r^(3)`

C

`(12)/sqrt(2)r^(3)`

D

`(64)/(3sqrt(3))r^(3)`

Text Solution

Verified by Experts

The correct Answer is:
1
Base area of regular hexagon=Area of 6 equilateral trinagles each with side 'a' `=6sqrt(3)/(4)a^(2)`
Height of the unit cell (h)=2xx distance between
adjacent layers `=2xxsqrt((2)/(3))a`
`therefore` Volume of the unit cell=Base area xx height
`==((6sqrt(3))/(4)a^(2))xx(2sqrt((2)/(3))a)=3sqrt(2)a^(3)`
Putting a=2 r, volume `=24sqrt(2)r^(3)`
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Knowledge Check

  • In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagon and three atoms are sandwiched in between them. A space filling model of this structure, called hexagonal close packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical sphere as closely as possible. Three spheres are then placed over the first layer so that they touch other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be same The volume of this HCP unit cell is

    A
    `24sqrt2r^(3)`
    B
    `16sqrt2r^(3)`
    C
    `12sqrt2r^(3)`
    D
    `64/(3sqrt3)r^(3)`

  • In hexagonal systems of crystals, a frequently ecounted arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are reqular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. There spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these three sphere touches three spheres of the bottom layer. Finally the second layer is covered with third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be 'r' The volume of this HCP unit cell is

    A
    `24sqrt2 r^(3)`
    B
    `16 sqrt2 r^(3)`
    C
    `12 sqrt2 r^(3)`
    D
    `(64)/(3 sqrt3)r^(3)`

  • In hexogonal systems of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexongonas and three atoms are sandwiched inbetween them. A space filling model of this structure called haxagonal closed packed (HCP) is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible . Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be 'r'. The volume of this HCP unit cell is

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    C
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    D
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