Show that, for Hexagonal Close-Packed Structure, c/a = 1.633, and Atomic Packing Factor (APF) = 0.74
The Correct Answer and Explanation is:
To show that for a Hexagonal Close-Packed (HCP) structure:
- c/a = 1.633, and
- Atomic Packing Factor (APF) = 0.74,
we analyze the ideal geometry of an HCP unit cell.
1. c/a Ratio Derivation
The HCP unit cell consists of 6 atoms and has an ABAB stacking sequence. The key is to consider the geometry formed by atoms in layers A and B.
- Let the atomic radius be R.
- In the hexagonal base (A layer), atoms touch each other, forming an equilateral triangle:
So, side a = 2R. - Now, consider the triangle formed by three atoms: one from the top A layer, one from the middle B layer, and one from the bottom A layer. The height between the two A layers is c, and we can analyze the tetrahedron formed by the 4 atoms: three in one plane and one directly above the center.
Using geometry: c=223R=2R83=2R⋅263c = 2 \sqrt{\frac{2}{3}} R = 2R \sqrt{\frac{8}{3}} = 2R \cdot \frac{2\sqrt{6}}{3}
So, the c/a ratio becomes: ca=2R⋅2632R=263≈1.633\frac{c}{a} = \frac{2R \cdot \frac{2\sqrt{6}}{3}}{2R} = \frac{2\sqrt{6}}{3} \approx 1.633
2. Atomic Packing Factor (APF)
APF is defined as the fraction of volume occupied by atoms in the unit cell: APF=Volume of atoms in unit cellVolume of unit cell\text{APF} = \frac{\text{Volume of atoms in unit cell}}{\text{Volume of unit cell}}
- Each unit cell has 6 atoms.
- Volume of one atom: V=43πR3V = \frac{4}{3} \pi R^3
- Volume of 6 atoms: 6⋅43πR3=8πR36 \cdot \frac{4}{3} \pi R^3 = 8\pi R^3
- Volume of unit cell: Vcell=Base Area×c=(332a2)cV_{cell} = \text{Base Area} \times c = \left(\frac{3\sqrt{3}}{2} a^2\right)c Substituting a=2Ra = 2R, and c=1.633⋅ac = 1.633 \cdot a: Vcell=332(2R)2(1.633⋅2R)=332⋅4R2⋅3.266R=63⋅3.266R3V_{cell} = \frac{3\sqrt{3}}{2} (2R)^2 (1.633 \cdot 2R) = \frac{3\sqrt{3}}{2} \cdot 4R^2 \cdot 3.266R = 6\sqrt{3} \cdot 3.266 R^3
Now calculate APF: APF=8πR363⋅3.266R3≈0.74\text{APF} = \frac{8\pi R^3}{6\sqrt{3} \cdot 3.266 R^3} \approx 0.74
Final Result:
- c/a = 1.633
- APF = 0.74
These values describe the ideal HCP structure, as shown in the slide.
