Select the correct statement(s) about three- dimensional hcp system.
This question has multiple correct options
a) The number of atoms in hcp unit cell is 6.
b) The volume of hcp unit cell is 24$\sqrt{2}$r$^{3}$.
c) The empty space in hcp unit cell is 26%.
d) The base area of hcp unit cell is 6$\sqrt{3}$r$^{2}$.

Hint: The three-dimensional hcp system is referred to the hexagonal close packing in the cubic lattice of crystals. From the name itself, we can determine the correct statement about the hcp system

Complete answer:
> First, let us discuss the three-dimensional hcp system. In this arrangement of atoms, the unit cell consists of three layers of atoms. The top layer will contain six atoms at the corners of the hexagon and one atom at the centre of each hexagon.
>If we talk about the number of atoms in an hcp unit cell.
12 atoms are present at corners, and each atom contributes one-sixth to the unit cell, as mentioned each corner atom is shared by 6 unit cells.
>As we know, 2 atoms are present at face centres, and each will contribute one half; and 3 atoms present in the body contributes 1 to the unit cell.
>Thus, we can write the number of atoms in hcp unit cell is
12$\times$ $frac{1}{6}$ (corners) + 2 $\times$ $frac{1}{2}$ (face centres) + 3 $\times$ $frac{1}{1}$ (in the body) = 2+1+3 = 6
>The next we have volume of unit cell = base area $\times$ height
>As we know that base area of regular hexagon = 6 $\times$ area of equilateral triangle = 6$\sqrt{3}$r$^{2}$.
>Now, the height of the unit cell = 4r$\sqrt{\dfrac{2}{3}}$.
>Thus, the volume of the unit cell is 6$\sqrt{3}$r$^{2}$4r$\sqrt{\dfrac{2}{3}}$ = 24$\sqrt{2}$r$^{3}$.
>We have calculated the volume of the unit cell by using the height, and the base area of the regular hexagon as mentioned above.
>According to the concept of lattice arrangement the empty space of atoms in the unit cell is 26%.
In the last, we can say that all the given statements are correct. The correct option is (A), (B), (C), and (D).

Note: Don’t get confused while calculating the volume of the unit cell. Just remember the hcp unit cell is hexagonal in shape. The hexagon is made up of 6 equilateral triangles. Thus, to calculate the base area, multiply the area of the equilateral triangle with 6.