Question

Ideal c/a ratio of HCP is

• A. 1.633
• B. 1.88
• C. 1.58
• D. 1.48

Hint:

Real HCP metals deviate slightly from the ideal ratio:
Zinc ($c/a \approx 1.856$) is stretched along the c-axis.
Titanium ($c/a \approx 1.587$) is compressed along the c-axis.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the value of the ideal $c/a$ ratio (height-to-edge ratio) of a Hexagonal Close-Packed (HCP) crystal structure.

Step 2: Key Formula or Approach:
In an ideal HCP structure composed of hard spheres of radius $R$ in contact, the parameters are:
$a = 2R$ (the base edge length).
$c$ is the height of the unit cell.
The ideal geometric ratio is calculated using a regular tetrahedron formed by three atoms in the middle layer and one atom in the top layer.

Step 3: Detailed Explanation:

• Consider the tetrahedron formed by four adjacent touching atoms in the HCP structure. The edge length of this tetrahedron is equal to the base lattice parameter $a$.

• The height of this regular tetrahedron is half the height of the unit cell ($c/2$).

• Using trigonometry on a regular tetrahedron of edge $a$:
\[ \left(\frac{c}{2}\right)^2 = a^2 - \left(\frac{a}{\sqrt{3}}\right)^2 \]
\[ \frac{c^2}{4} = a^2 - \frac{a^2}{3} = \frac{2}{3}a^2 \]
\[ c^2 = \frac{8}{3}a^2 \]
\[ \frac{c}{a} = \sqrt{\frac{8}{3}} \approx 1.633 \]

• This ratio of $1.633$ yields the maximum atomic packing factor of $0.74$, matching the packing density of the FCC structure.

Step 4: Final Answer:
The ideal $c/a$ ratio of HCP is $1.633$.