Consider a rectangular lattice in two dimensions with primitive lattice vectors $(a,0)$ and $(0,2a)$.
Which of the following are reciprocal lattice vectors for this lattice?
(a) $\quad\dfrac{\pi}{a}\left(1,\frac12 \right)$
(b) $\quad\dfrac{\pi}{a}\left(1,2 \right)$
(c) $\quad\dfrac{\pi}{a}\left(2,-1 \right)$
(d) $\quad\dfrac{\pi}{a}\left(0,2 \right)$
(e) $\quad\dfrac{\pi}{a}\left(\frac12,-2 \right)$
My attempt is:
Let the reciprocal lattice vectors $\vec b_1$ & $\vec b_2$ be
$$\vec b_1=\begin{bmatrix}x_1\\ y_1\\ \end{bmatrix}\qquad \text{and}\qquad\vec b_2=\begin{bmatrix}x_2\\ y_2\\ \end{bmatrix}$$ respectively.
and denote the primitive lattice vectors $\vec a_1$ & $\vec a_2$ as
$$\vec a_1=\begin{bmatrix}a\\ 0\\ \end{bmatrix}\qquad \text{and}\qquad\vec a_2=\begin{bmatrix}0\\ 2a\\ \end{bmatrix}$$
respectively
Now using the condition, $\vec a_i \cdot \vec b_j=2\pi\delta_{ij}\tag{1}$
$\vec a_1 \cdot \vec b_1=\begin{bmatrix}a\\ 0\\ \end{bmatrix}\cdot\begin{bmatrix}x_1\\ y_1\\ \end{bmatrix}= 2\pi$
So $x_1 =\frac{2\pi}{a}$
$\vec a_2 \cdot \vec b_1=\begin{bmatrix}0\\ 2a\\ \end{bmatrix}\cdot\begin{bmatrix}\frac{2\pi}{a}\\ y_1\\ \end{bmatrix}= 0\implies$ $y_1=0$
and hence
$\vec b_1 = \begin{bmatrix}\frac{2\pi}{a}\\ 0\\ \end{bmatrix}=\color{red}{\frac{\pi}{a}\left(2,0\right)}$
Now to find $\vec b_2$,
$\vec a_2 \cdot \vec b_2=\begin{bmatrix}0\\ 2a\\ \end{bmatrix}\cdot\begin{bmatrix}x_2\\ y_2\\ \end{bmatrix}= 2\pi\implies y_2 = \frac{\pi}{a}$
$\vec a_1 \cdot \vec b_2=\begin{bmatrix}a\\ 0\\ \end{bmatrix}\cdot\begin{bmatrix}x_2\\ \frac{\pi}{a}\\ \end{bmatrix}= 0\implies x_2 = 0$ and hence $\vec b_2= \begin{bmatrix}0\\ \frac{\pi}{a}\\ \end{bmatrix}=\color{red}{\frac{\pi}{a}\left(0,1\right)}$
The two vectors calculated (in red) do not correspond to any of the answers above, from which I can tell you that 2 of them are correct. I thought that I was applying the condition $(1)$ correctly.
Could someone please explain to me what I'm doing wrong?
Best Answer
I have just realised that the vectors calculated $\vec b_1=\color{red}{\frac{\pi}{a}\left(2,0\right)}$ & $\vec b_2=\color{red}{\frac{\pi}{a}\left(0,1\right)}$ are still correct answers but they are also the primitive reciprocal lattice vectors.
Any integer multiple of the primitive reciprocal lattice vectors is itself another reciprocal lattice vector:
$2 \vec b_2=\frac{\pi}{a}\left(0,2\right)$ and $\vec b_1 - \vec b_2 = \frac{\pi}{a}\left(2,-1\right)$
So the correct answers are $(\rm{c})$ and $(\rm{d})$