Understanding the Ideal HCP Structure Ratio

The question asks for the ratio $\frac{c}{a}$ in an ideal hexagonal close-packed (hcp) structure. An ideal hcp structure assumes atoms are hard spheres that touch each other, and specific geometric relationships hold.

Deriving the c/a Ratio

1. Relationship between lattice parameters and atomic radius (R):

   • In the basal plane (the 'a' dimension), three adjacent atoms touch each other, forming an equilateral triangle. The side length of this triangle is the distance between the centers of two touching atoms, which is $2R$. Thus, the lattice parameter 'a' is equal to $2R$.

     $a = 2R$

2. Calculating the height (c):

   • Consider the vertical distance. An ideal hcp structure can be thought of layers stacked in an ABABA... sequence. The distance 'c' represents the height of the unit cell.
   • Focus on the vertical distance ('h') between the centers of atoms in adjacent layers (e.g., layer A and layer B). This distance can be found by considering a tetrahedron formed by three atoms in layer A and one atom in layer B situated above the hollow of the layer A atoms.
   • The edges connecting atoms between layers also have length $2R$. The distance from the center of the basal triangle (formed by 3 atoms in layer A) to one of its vertices is $\frac{a}{\sqrt{3}}$.
   • Using the Pythagorean theorem on the right triangle formed by the apex atom (layer B), the center of the basal triangle (layer A), and a vertex atom (layer A):

     $ (2R)^2 = h^2 + \left(\frac{a}{\sqrt{3}}\right)^2 $

   • Substitute $a = 2R$:

     $ a^2 = h^2 + \left(\frac{a}{\sqrt{3}}\right)^2 $

     $ a^2 = h^2 + \frac{a^2}{3} $

   • Solve for $h^2$:

     $ h^2 = a^2 - \frac{a^2}{3} = \frac{2a^2}{3} $

     $ h = a\sqrt{\frac{2}{3}} $

   • The total height 'c' of the unit cell is twice the distance 'h' between layers because the unit cell contains two such layer spacings contributing to the height.

     $ c = 2h = 2a\sqrt{\frac{2}{3}} $

3. Calculating the c/a Ratio:

   • Divide the expression for 'c' by 'a':

     $ \frac{c}{a} = \frac{2a\sqrt{\frac{2}{3}}}{a} = 2\sqrt{\frac{2}{3}} $

   • Simplify the expression:

     $ \frac{c}{a} = \sqrt{4 \times \frac{2}{3}} = \sqrt{\frac{8}{3}} $

   • Calculate the numerical value:

     $ \frac{c}{a} \approx \sqrt{2.666...} \approx 1.633 $

Therefore, the ratio $\frac{c}{a}$ for an ideal hcp structure is approximately 1.633.