Question

The direct lattices are given by \(a_1=(\hat{i}+\hat{j}+\hat{k})\), \(a_2=(3\hat{i}-2\hat{k})\), and \(a_3=(4\hat{i}+3\hat{j})\). Find out the reciprocal lattices \(b_1\), \(b_2\), and \(b_3\).

• A. \(b_1=2\pi(3\hat{i}-6\hat{j}+3\hat{k}),\ b_2=2\pi(3\hat{i}-\hat{j}+\hat{k}),\ b_3=2\pi(2\hat{i}-\hat{j}+\hat{k})\)
• B. \(b_1=2\pi(6\hat{i}-8\hat{j}+9\hat{k}),\ b_2=2\pi(3\hat{i}-4\hat{j}+\hat{k}),\ b_3=2\pi(-2\hat{i}+5\hat{j}-3\hat{k})\)
• C. \(b_1=2\pi(6\hat{i}-8\hat{j}+9\hat{k}),\ b_2=2\pi(2\hat{i}-3\hat{j}+3\hat{k}),\ b_3=2\pi(2\hat{i}-\hat{j}+\hat{k})\)
• D. \(b_1=2\pi(3\hat{i}-6\hat{j}+3\hat{k}),\ b_2=2\pi(3\hat{i}-4\hat{j}+4\hat{k}),\ b_3=2\pi(2\hat{i}-\hat{j}+\hat{k})\)

Hint:

Reciprocal lattice vectors are obtained using cross products of the direct lattice vectors.

Answer 1

The Correct Option is B

Concept:
For direct lattice vectors \(a_1,a_2,a_3\), reciprocal lattice vectors are: \[ b_1=2\pi\frac{a_2\times a_3}{a_1\cdot(a_2\times a_3)} \] \[ b_2=2\pi\frac{a_3\times a_1}{a_1\cdot(a_2\times a_3)} \] \[ b_3=2\pi\frac{a_1\times a_2}{a_1\cdot(a_2\times a_3)} \]

Step 1: Write the given vectors. \[ a_1=(1,1,1),\quad a_2=(3,0,-2),\quad a_3=(4,3,0) \]

Step 2: Find cross products. \[ a_2\times a_3=(6,-8,9) \] \[ a_3\times a_1=(3,-4,1) \] \[ a_1\times a_2=(-2,5,-3) \]

Step 3: Write reciprocal lattice vectors. \[ b_1=2\pi(6\hat{i}-8\hat{j}+9\hat{k}) \] \[ b_2=2\pi(3\hat{i}-4\hat{j}+\hat{k}) \] \[ b_3=2\pi(-2\hat{i}+5\hat{j}-3\hat{k}) \] \[ \therefore \text{Correct Answer is (B)} \]