In hexagonal 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 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. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these 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 'r'. The number of atoms in this hcp unit cell is; The volume of this hcp unit cell is: The empty space in this hcp unit cell is:

Question

In hexagonal 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 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. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these 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 'r'. The number of atoms in this hcp unit cell is; The volume of this hcp unit cell is: The empty space in this hcp unit cell is:

Infinity Learn · Accepted Answer

Total no. of atoms in $$1$$ unit cell of hcp $$=(12$$×$$1/6)+3+(2$$×$$1/2)=6$$  Volume of unit cell $$=$$ Height × Base area                     $$=4r23$$×$$6$$×$$34$$×$$4r2=242r3$$∵ Packing $$fraction=$$ Volume of the atoms in one unit cell  Volume of one unit cell $$=6$$×$$43$$$$π$$$$r3242r3=$$π$$32=0.74=74$$% ∴Empty space $$=$$ $$26$$%

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