Question

Which of these cubic lattice plane pairs is(are) perpendicular to each other?

• A. (100), (010)
• B. (220), (001)
• C. (110), (010)
• D. (112), (220)

Hint:

This method of using the dot product of normal vectors works specifically for cubic lattices because the crystal axes are orthogonal. For other crystal systems (tetragonal, orthorhombic, etc.), the formula for the angle between planes is more complex as it involves the lattice parameters.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In crystallography, a lattice plane is represented by its Miller indices (hkl). The direction perpendicular to the (hkl) plane in a cubic lattice is given by the vector \([hkl]\), which is \(h\hat{i} + k\hat{j} + l\hat{k}\). Two planes are perpendicular to each other if the vectors normal to them are perpendicular.
Step 2: Key Formula or Approach:
Two vectors \(\mathbf{v}_1 = h_1\hat{i} + k_1\hat{j} + l_1\hat{k}\) and \(\mathbf{v}_2 = h_2\hat{i} + k_2\hat{j} + l_2\hat{k}\) are perpendicular if their dot product is zero.
\[ \mathbf{v}_1 \cdot \mathbf{v}_2 = h_1h_2 + k_1k_2 + l_1l_2 = 0 \] We need to apply this condition to the normal vectors of the given pairs of planes.
Step 3: Detailed Explanation:
Let's check each pair:
(A) (100) and (010)
The normal vectors are \(\mathbf{v}_1 = 1\hat{i} + 0\hat{j} + 0\hat{k}\) and \(\mathbf{v}_2 = 0\hat{i} + 1\hat{j} + 0\hat{k}\).
Dot product: \(\mathbf{v}_1 \cdot \mathbf{v}_2 = (1)(0) + (0)(1) + (0)(0) = 0\).
Since the dot product is 0, the planes are perpendicular. So, (A) is correct.
(B) (220) and (001)
The normal vectors are \(\mathbf{v}_1 = 2\hat{i} + 2\hat{j} + 0\hat{k}\) and \(\mathbf{v}_2 = 0\hat{i} + 0\hat{j} + 1\hat{k}\).
Dot product: \(\mathbf{v}_1 \cdot \mathbf{v}_2 = (2)(0) + (2)(0) + (0)(1) = 0\).
Since the dot product is 0, the planes are perpendicular. So, (B) is correct.
(C) (110) and (010)
The normal vectors are \(\mathbf{v}_1 = 1\hat{i} + 1\hat{j} + 0\hat{k}\) and \(\mathbf{v}_2 = 0\hat{i} + 1\hat{j} + 0\hat{k}\).
Dot product: \(\mathbf{v}_1 \cdot \mathbf{v}_2 = (1)(0) + (1)(1) + (0)(0) = 1\).
Since the dot product is not 0, the planes are not perpendicular. So, (C) is incorrect.
(D) (112) and (220)
The normal vectors are \(\mathbf{v}_1 = 1\hat{i} + 1\hat{j} + 2\hat{k}\) and \(\mathbf{v}_2 = 2\hat{i} + 2\hat{j} + 0\hat{k}\).
Dot product: \(\mathbf{v}_1 \cdot \mathbf{v}_2 = (1)(2) + (1)(2) + (2)(0) = 2 + 2 + 0 = 4\).
Since the dot product is not 0, the planes are not perpendicular. So, (D) is incorrect.
Step 4: Final Answer:
The pairs of planes that are perpendicular to each other are (100), (010) and (220), (001). This corresponds to options (A) and (B).