Question

The primitive translation vectors of a two-dimensional lattice are \(\vec{a} = 2\hat{i} + \hat{j}\), \(\vec{b} = 2\hat{j}\). The primitive translation vector of its reciprocal lattice in the \(x\)-direction is given by:

• A. \(\vec{a}^{*} = 2\pi\hat{i}\)
• B. \(\vec{a}^{*} = \pi\hat{i}\)
• C. \(\vec{a}^{*} = 3\pi\hat{i}\)
• D. \(\vec{a}^{*} = \pi\hat{j}\)

Hint:

Require \(\vec{a}^{*}\cdot\vec{b}=0\) (forces it along \(x\)) and \(\vec{a}^{*}\cdot\vec{a}=2\pi\).

Answer 1

The Correct Option is B

Step 1: The reciprocal vectors satisfy \(\vec{a}^{*}\cdot\vec{a}=2\pi\) and \(\vec{a}^{*}\cdot\vec{b}=0\).

Step 2: Use the 2D formula with \(\hat{n}=\hat{k}\):
\[\vec{a}^{*} = 2\pi\,\frac{\vec{b}\times\hat{n}}{\vec{a}\cdot(\vec{b}\times\hat{n})}.\]
Step 3: Compute \(\vec{b}\times\hat{n} = (2\hat{j})\times\hat{k} = 2\hat{i}\).

Step 4: Denominator: \(\vec{a}\cdot(2\hat{i}) = (2\hat{i}+\hat{j})\cdot 2\hat{i} = 4\).

Step 5: Therefore
\[\vec{a}^{*} = 2\pi\,\frac{2\hat{i}}{4} = \pi\hat{i}.\]
\[\boxed{\vec{a}^{*} = \pi\hat{i}}\]