For a given crystal the lattice parameter a is 318 pm. The d-spacing for a (III) plane is - (a) 225 pm (b) 184 pm (c) 318 pm (d) 390 pm

To determine the d-spacing for a specific crystal plane, we can use Bragg's Law and the relationship between the lattice parameter and the d-spacing. In this case, we are looking for the d-spacing for the (III) plane of a crystal with a lattice parameter \( a \) of 318 pm. Let's break this down step by step.

Understanding Lattice Parameters and d-Spacing

The lattice parameter \( a \) is the distance between two adjacent points in the crystal lattice. The d-spacing, or interplanar spacing, is the distance between two parallel planes in the crystal structure. For cubic crystals, the relationship between the lattice parameter and the d-spacing can be expressed using the formula:

d = a / √(h² + k² + l²)

Here, \( h \), \( k \), and \( l \) are the Miller indices of the crystal plane. For the (III) plane, the Miller indices are \( h = 3 \), \( k = 0 \), and \( l = 0 \).

Calculating the d-Spacing

Now, let's plug in the values into the formula:

• Given \( a = 318 \) pm
• Miller indices for (III) plane: \( h = 3 \), \( k = 0 \), \( l = 0 \)
• Calculate \( h² + k² + l² = 3² + 0² + 0² = 9 \)

Now substituting into the d-spacing formula:

d = 318 pm / √(9)

Calculating the square root:

√(9) = 3

Now, substituting back into the equation:

d = 318 pm / 3 = 106 pm

Evaluating the Options

Now that we have calculated the d-spacing for the (III) plane, we can compare it to the options provided:

• (a) 225 pm
• (b) 184 pm
• (c) 318 pm
• (d) 390 pm

None of the options match our calculated d-spacing of 106 pm. This suggests that either the question may have a typo or the options provided do not correspond to the (III) plane as calculated. It’s essential to double-check the parameters or the plane indices if you encounter such discrepancies.

Final Thoughts

Understanding the relationship between lattice parameters and d-spacing is crucial in crystallography. It helps in determining how atoms are arranged in a crystal and can influence properties such as diffraction patterns. If you have any further questions or need clarification on any part of this process, feel free to ask!