Calculate the distance between 111 planes in a crystal of Ca. Options are: A. 1.61nm B. 0.61nm C. 0.321nm D. None of these.

To calculate the distance between the 111 planes in a crystal of calcium (Ca), we can use the formula derived from Bragg's Law and the concept of Miller indices. The distance between planes in a crystal lattice is often referred to as the interplanar spacing, denoted as d. For cubic crystals, the formula for the interplanar spacing is given by:

Interplanar Spacing Formula

The formula for the interplanar spacing d for a cubic crystal system is:

d = a / √(h² + k² + l²)

Where:

• d = interplanar spacing
• a = lattice parameter (the edge length of the unit cell)
• h, k, l = Miller indices of the plane

Step-by-Step Calculation

For the 111 planes, the Miller indices are (1, 1, 1). Thus, we substitute h = 1, k = 1, and l = 1 into the formula:

d = a / √(1² + 1² + 1²)

This simplifies to:

d = a / √3

Finding the Lattice Parameter

Next, we need the lattice parameter (a) for calcium. The lattice parameter for calcium in its face-centered cubic (FCC) structure is approximately 0.558 nm. Now we can substitute this value into our equation:

d = 0.558 nm / √3

Calculating this gives:

d ≈ 0.558 nm / 1.732

d ≈ 0.321 nm

Final Answer

Thus, the distance between the 111 planes in a crystal of calcium is approximately 0.321 nm. Therefore, the correct option is C. 0.321nm.