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: Impose the defining conditions directly. Let $\vec{a}^{*}=(p,q)$ in the $(\hat{i},\hat{j})$ basis.

Step 2: Orthogonality to $\vec{b}=2\hat{j}$: $\vec{a}^{*}\cdot\vec{b}=2q=0$, so $q=0$. Hence $\vec{a}^{*}$ points purely along $x$.

Step 3: Normalization with $\vec{a}=2\hat{i}+\hat{j}$: $\vec{a}^{*}\cdot\vec{a}=2p+q=2p=2\pi$.

Step 4: Solve $2p=2\pi\Rightarrow p=\pi$.

Step 5: Assemble the vector.
\[\boxed{\vec{a}^{*}=\pi\hat{i}}\]