**Step-(1)**In order to develop

**three dimensional close packing**take a 2D hexagonal close packing sheet as first layer

**(A- layer**).

**Step-(2)**Another 2D hexagonal close sheet

**(B-layer)**is taken and it is just over the depression (Pit) of the first layer (A) .When the second layer is placed in such a way that its spheres find place in the

**‘b’ voids**of the first layer, the

**‘c’ voids**will be left unoccupied. Since under this arrangement no sphere can be placed in them, (c voids), i.e. only half (50%) the triangular voids in the first layer are occupied by spheres of the second layer (i.e. either b or c)

**Step-(3)**There are two alternative ways in which spheres in the third layer can be arranged over the second layer

**(1)**When a third layer is placed over the second layer in such a way that the spheres cover the

**tetrahedral or ‘a’ voids;**a three dimensional closest packing is obtained where the spheres in every third or alternate layers are vertically aligned (i.e. the third layer is directly above the first, the fourth above the second layer and so on) calling the first layer A and second layer as layer B,

**the arrangement is called A**BAB …………. pattern or

**hexagonal close packing (HCP)**as it has hexagonal symmetry.

**ANALYSIS OF HCP UNIT CELL:**

**(1) Number of effective atoms in HCP unit cell (Z):**

**Lattice point: corner-**total 12 carbon contribute 1/6 to the unit cell

**Lattice point: face-**total face 2 contribute ½

**Lattice point: body centre-**total atom 3 (100% contribution)

**(2) Radius of atom in HCP unit cell:**

Let the edge of
hexagonal base

**=( a)**And the height of hexagon =**( h)**And radius of sphere**=( r)****(3) Area of hexagon:**

**Area of hexagonal can be divided into**

**six**equilateral triangles with side

**2r**

**(4) Height of HCP unit cell:**

**(5)**

**Volume of hexagon = area of base x height**

**(6)**

**Volume of spheres:**

**(7) Packing efficiency**: Percentage of space occupied by sphere.

**(8) Voids %: 100 - PE= 26 %**

**(9) Coordination Numbers:**

**12 (**each spheres touches 6 spheres in its layer,3 above and 3 below)

**.**

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